Lagrange Multipliers Notes

Published on: September 29, 2026

This Lagrange Multipliers Calculator finds the largest and smallest values a function can take while a constraint is obeyed, and shows the working line by line. At such a point the two gradients line up, because otherwise you could still slide along the constraint and improve. Setting one gradient to a multiple of the other gives the equations to solve. Following the same pattern every time keeps the constraint in play throughout.

Step-by-step method

Set up the problem.
Find the gradients of f and g.
Write the Lagrange system grad f = lambda grad g together with the constraint.
Solve the system for x, y, and lambda.
Evaluate f at each solution and pick the largest and smallest.

Formula:

\(\nabla f = \lambda \nabla g, \quad g(x,y) = k\)

See Example 1 Hide Example 1

Problem

\(f = x y,\; g = x^{2} + y^{2} = 1\)

Approach

Step-by-step method

Set up the problem.
Find the gradients of f and g.
Write the Lagrange system grad f = lambda grad g together with the constraint.
Solve the system for x, y, and lambda.
Evaluate f at each solution and pick the largest and smallest.

Step 1

Step 1 - Set up the problem.

In this problem:

\(f = x y,\; g = x^{2} + y^{2} = 1\)

Step 2

Step 2 - Find the gradients of f and g.

In this problem:

\(\nabla f = \langle y,x \rangle,\quad \nabla g = \langle 2 x,2 y \rangle\)

Step 1

Step 1 - Set up the coefficient and the variable part.

In this problem: We are given \(x y\). The constant coefficient is \(y\), and the variable part is \(x\).

\(x y = y\cdot \left(x\right)\)

Step 2

Step 2 - Write the constant multiple rule formula.

In this problem: The constant multiple rule says a constant coefficient stays outside the derivative.

\(\frac{d}{dx}\left(c\cdot f\left(x\right)\right) = c\cdot \frac{d}{dx}f\left(x\right)\)

Step 3

Step 3 - Apply the constant multiple rule.

In this problem: Move the constant coefficient \(y\) outside the derivative and leave \(x\) inside the derivative.

\(\frac{d}{dx}\left(x y\right) = y\cdot \frac{d}{dx}\left(x\right)\)

Step 1

Step 1 - Set up the coefficient and the variable part.

In this problem: We are given \(x y\). The constant coefficient is \(x\), and the variable part is \(y\).

\(x y = x\cdot \left(y\right)\)

Step 2

Step 2 - Write the constant multiple rule formula.

In this problem: The constant multiple rule says a constant coefficient stays outside the derivative.

\(\frac{d}{dy}\left(c\cdot f\left(y\right)\right) = c\cdot \frac{d}{dy}f\left(y\right)\)

Step 3

Step 3 - Apply the constant multiple rule.

In this problem: Move the constant coefficient \(x\) outside the derivative and leave \(y\) inside the derivative.

\(\frac{d}{dy}\left(x y\right) = x\cdot \frac{d}{dy}\left(y\right)\)

Step 1

Step 1 - Set up the terms in the sum.

In this problem: We are given a sum. The separate terms are \(x^{2}\), \(y^{2}\).

\(f\left(x\right) = x^{2} + y^{2}\)

Step 2

Step 2 - Write the sum rule formula.

In this problem: The sum rule says the derivative of a sum is the sum of the derivatives.

\(\frac{d}{dx}\left(f\left(x\right) + g\left(x\right)\right) = \frac{d}{dx}f\left(x\right) + \frac{d}{dx}g\left(x\right)\)

Step 3

Step 3 - Split the derivative across the addition signs.

In this problem: Apply the derivative to each separated term.

\(\frac{d}{dx}\left(x^{2} + y^{2}\right) = \frac{d}{dx}\left(x^{2}\right) + \frac{d}{dx}\left(y^{2}\right)\)

Step 4a

Step 4a - Differentiate the separated term \(x^{2}\).

In this problem: The separated term is \(x^{2}\). Use the power rule to get its derivative.

\(\frac{d}{dx}\left(x^{2}\right) = 2 x\)

Problem

\(f\left(x\right) = {x}^{2}\)

Approach

Step-by-step method

Set up the problem.
If a constant coefficient is present, apply the constant multiple rule first.
Write the power rule formula.
Substitute the exponent into the power rule.
Solve and simplify.

Step 1

Step 1 - Set up the problem.

In this problem: We are given \(x^{2}\). This is a power of \(x\), so the exponent is \(n = 2\).

\(f\left(x\right) = {x}^{2}\)

Step 2

Step 2 - Write the power rule formula.

In this problem: The power rule says to move the exponent to the front, then subtract 1 from the exponent.

\(\frac{d}{dx}{x}^{n} = n\,{x}^{n - 1}\)

Step 3

Step 3 - Substitute the given power into the formula.

In this problem: Here, \(n = 2\). Substitute this exponent into the power rule, but do not simplify yet.

\(\frac{d}{dx}{x}^{2} = 2{x}^{2 - 1}\)

Step 4

Step 4 - Solve and simplify.

In this problem: Now subtract 1 from the exponent and keep the result in power form. This gives \(2{x}^{1}\).

\(2{x}^{2 - 1} = 2{x}^{1} = 2 x\)

Final Answer

\(2 x\)

Step 4b

Step 4b - Differentiate the separated term \(y^{2}\).

In this problem: The separated term is \(y^{2}\). Use the constant rule to get its derivative.

\(\frac{d}{dx}\left(y^{2}\right) = 0\)

Problem

\(f\left(x\right) = y^{2}\)

Approach

Step-by-step method

Set up the problem.
Write the constant rule formula.
Apply the constant rule.

Step 1

Step 1 - Set up the problem.

In this problem: We are given \(y^{2}\). This expression does not contain \(x\), so it is a constant.

\(f\left(x\right) = y^{2}\)

Step 2

Step 2 - Write the constant rule formula.

In this problem: The constant rule says the derivative of any constant is \(0\).

\(\frac{d}{dx}c = 0\)

Step 3

Step 3 - Apply the constant rule.

In this problem: Since \(y^{2}\) is constant, its derivative is \(0\).

\(\frac{d}{dx}\left(y^{2}\right) = 0\)

Final Answer

\(0\)

Step 5

Step 5 - Combine and simplify.

In this problem: Combine the derivative from each separated term, then simplify the result.

\(f'\left(x\right) = 2 x + 0 = 2 x\)

Step 1

Step 1 - Set up the terms in the sum.

In this problem: We are given a sum. The separate terms are \(x^{2}\), \(y^{2}\).

\(f\left(y\right) = x^{2} + y^{2}\)

Step 2

Step 2 - Write the sum rule formula.

In this problem: The sum rule says the derivative of a sum is the sum of the derivatives.

\(\frac{d}{dy}\left(f\left(y\right) + g\left(y\right)\right) = \frac{d}{dy}f\left(y\right) + \frac{d}{dy}g\left(y\right)\)

Step 3

Step 3 - Split the derivative across the addition signs.

In this problem: Apply the derivative to each separated term.

\(\frac{d}{dy}\left(x^{2} + y^{2}\right) = \frac{d}{dy}\left(x^{2}\right) + \frac{d}{dy}\left(y^{2}\right)\)

Step 4a

Step 4a - Differentiate the separated term \(x^{2}\).

In this problem: The separated term is \(x^{2}\). Use the constant rule to get its derivative.

\(\frac{d}{dy}\left(x^{2}\right) = 0\)

Problem

\(f\left(y\right) = x^{2}\)

Approach

Step-by-step method

Set up the problem.
Write the constant rule formula.
Apply the constant rule.

Step 1

Step 1 - Set up the problem.

In this problem: We are given \(x^{2}\). This expression does not contain \(y\), so it is a constant.

\(f\left(y\right) = x^{2}\)

Step 2

Step 2 - Write the constant rule formula.

In this problem: The constant rule says the derivative of any constant is \(0\).

\(\frac{d}{dx}c = 0\)

Step 3

Step 3 - Apply the constant rule.

In this problem: Since \(x^{2}\) is constant, its derivative is \(0\).

\(\frac{d}{dy}\left(x^{2}\right) = 0\)

Final Answer

\(0\)

Step 4b

Step 4b - Differentiate the separated term \(y^{2}\).

In this problem: The separated term is \(y^{2}\). Use the power rule to get its derivative.

\(\frac{d}{dy}\left(y^{2}\right) = 2 y\)

Problem

\(f\left(y\right) = {y}^{2}\)

Approach

Step-by-step method

Set up the problem.
If a constant coefficient is present, apply the constant multiple rule first.
Write the power rule formula.
Substitute the exponent into the power rule.
Solve and simplify.

Step 1

Step 1 - Set up the problem.

In this problem: We are given \(y^{2}\). This is a power of \(x\), so the exponent is \(n = 2\).

\(f\left(y\right) = {y}^{2}\)

Step 2

Step 2 - Write the power rule formula.

In this problem: The power rule says to move the exponent to the front, then subtract 1 from the exponent.

\(\frac{d}{dy}{y}^{n} = n\,{y}^{n - 1}\)

Step 3

Step 3 - Substitute the given power into the formula.

In this problem: Here, \(n = 2\). Substitute this exponent into the power rule, but do not simplify yet.

\(\frac{d}{dy}{y}^{2} = 2{y}^{2 - 1}\)

Step 4

Step 4 - Solve and simplify.

In this problem: Now subtract 1 from the exponent and keep the result in power form. This gives \(2{y}^{1}\).

\(2{y}^{2 - 1} = 2{y}^{1} = 2 y\)

Final Answer

\(2 y\)

Step 5

Step 5 - Combine and simplify.

In this problem: Combine the derivative from each separated term, then simplify the result.

\(f'\left(y\right) = 0 + 2 y = 2 y\)

Step 3

Step 3 - Write the Lagrange system grad f = lambda grad g together with the constraint.

In this problem:

\(\begin{cases} y = \lambda\left(2 x\right) \\ x = \lambda\left(2 y\right) \\ x^{2} + y^{2} = 1 \end{cases}\)

Step 4

Step 4 - Solve the system for x, y, and lambda.

In this problem:

\(\begin{gathered} f(- \frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}) = - \frac{1}{2} \\ f(\frac{\sqrt{2}}{2},- \frac{\sqrt{2}}{2}) = - \frac{1}{2} \\ f(- \frac{\sqrt{2}}{2},- \frac{\sqrt{2}}{2}) = \frac{1}{2} \\ f(\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}) = \frac{1}{2} \end{gathered}\)

Step 5

Step 5 - Evaluate f at each solution and pick the largest and smallest.

In this problem:

\(f_{\max} = \frac{1}{2},\quad f_{\min} = - \frac{1}{2}\)

Final Answer

\(f_{\max} = \frac{1}{2},\quad f_{\min} = - \frac{1}{2}\)

See Example 2 Hide Example 2

Problem

\(f = x + y,\; g = x^{2} + y^{2} = 2\)

Approach

Step-by-step method

Set up the problem.
Find the gradients of f and g.
Write the Lagrange system grad f = lambda grad g together with the constraint.
Solve the system for x, y, and lambda.
Evaluate f at each solution and pick the largest and smallest.

Step 1

Step 1 - Set up the problem.

In this problem:

\(f = x + y,\; g = x^{2} + y^{2} = 2\)

Step 2

Step 2 - Find the gradients of f and g.

In this problem:

\(\nabla f = \langle 1,1 \rangle,\quad \nabla g = \langle 2 x,2 y \rangle\)

Step 1

Step 1 - Set up the terms in the sum.

In this problem: We are given a sum. The separate terms are \(x\), \(y\).

\(f\left(x\right) = x + y\)

Step 2

Step 2 - Write the sum rule formula.

In this problem: The sum rule says the derivative of a sum is the sum of the derivatives.

\(\frac{d}{dx}\left(f\left(x\right) + g\left(x\right)\right) = \frac{d}{dx}f\left(x\right) + \frac{d}{dx}g\left(x\right)\)

Step 3

Step 3 - Split the derivative across the addition signs.

In this problem: Apply the derivative to each separated term.

\(\frac{d}{dx}\left(x + y\right) = \frac{d}{dx}\left(x\right) + \frac{d}{dx}\left(y\right)\)

Step 4a

Step 4a - Differentiate the separated term \(x\).

In this problem: The separated term is \(x\). Use the variable rule to get its derivative.

\(\frac{d}{dx}\left(x\right) = 1\)

Problem

\(f\left(x\right) = x\)

Approach

Step-by-step method

Set up the problem.
Write the variable rule formula.
Apply the variable rule.

Step 1

Step 1 - Set up the problem.

In this problem: We are given \(x\). This is just the variable \(x\).

\(f\left(x\right) = x\)

Step 2

Step 2 - Write the variable rule formula.

In this problem: The variable rule says the derivative of \(x\) with respect to \(x\) is \(1\).

\(\frac{d}{dx}x = 1\)

Step 3

Step 3 - Apply the variable rule.

In this problem: Since the expression is \(x\), its derivative is \(1\).

\(\frac{d}{dx}x = 1\)

Final Answer

\(1\)

Step 4b

Step 4b - Differentiate the separated term \(y\).

In this problem: The separated term is \(y\). Use the constant rule to get its derivative.

\(\frac{d}{dx}\left(y\right) = 0\)

Problem

\(f\left(x\right) = y\)

Approach

Step-by-step method

Set up the problem.
Write the constant rule formula.
Apply the constant rule.

Step 1

Step 1 - Set up the problem.

In this problem: We are given \(y\). This expression does not contain \(x\), so it is a constant.

\(f\left(x\right) = y\)

Step 2

Step 2 - Write the constant rule formula.

In this problem: The constant rule says the derivative of any constant is \(0\).

\(\frac{d}{dx}c = 0\)

Step 3

Step 3 - Apply the constant rule.

In this problem: Since \(y\) is constant, its derivative is \(0\).

\(\frac{d}{dx}\left(y\right) = 0\)

Final Answer

\(0\)

Step 5

Step 5 - Combine and simplify.

In this problem: Combine the derivative from each separated term, then simplify the result.

\(f'\left(x\right) = 1 + 0 = 1\)

Step 1

Step 1 - Set up the terms in the sum.

In this problem: We are given a sum. The separate terms are \(x\), \(y\).

\(f\left(y\right) = x + y\)

Step 2

Step 2 - Write the sum rule formula.

In this problem: The sum rule says the derivative of a sum is the sum of the derivatives.

\(\frac{d}{dy}\left(f\left(y\right) + g\left(y\right)\right) = \frac{d}{dy}f\left(y\right) + \frac{d}{dy}g\left(y\right)\)

Step 3

Step 3 - Split the derivative across the addition signs.

In this problem: Apply the derivative to each separated term.

\(\frac{d}{dy}\left(x + y\right) = \frac{d}{dy}\left(x\right) + \frac{d}{dy}\left(y\right)\)

Step 4a

Step 4a - Differentiate the separated term \(x\).

In this problem: The separated term is \(x\). Use the constant rule to get its derivative.

\(\frac{d}{dy}\left(x\right) = 0\)

Problem

\(f\left(y\right) = x\)

Approach

Step-by-step method

Set up the problem.
Write the constant rule formula.
Apply the constant rule.

Step 1

Step 1 - Set up the problem.

In this problem: We are given \(x\). This expression does not contain \(y\), so it is a constant.

\(f\left(y\right) = x\)

Step 2

Step 2 - Write the constant rule formula.

In this problem: The constant rule says the derivative of any constant is \(0\).

\(\frac{d}{dx}c = 0\)

Step 3

Step 3 - Apply the constant rule.

In this problem: Since \(x\) is constant, its derivative is \(0\).

\(\frac{d}{dy}\left(x\right) = 0\)

Final Answer

\(0\)

Step 4b

Step 4b - Differentiate the separated term \(y\).

In this problem: The separated term is \(y\). Use the variable rule to get its derivative.

\(\frac{d}{dy}\left(y\right) = 1\)

Problem

\(f\left(y\right) = y\)

Approach

Step-by-step method

Set up the problem.
Write the variable rule formula.
Apply the variable rule.

Step 1

Step 1 - Set up the problem.

In this problem: We are given \(y\). This is just the variable \(y\).

\(f\left(y\right) = y\)

Step 2

Step 2 - Write the variable rule formula.

In this problem: The variable rule says the derivative of \(y\) with respect to \(y\) is \(1\).

\(\frac{d}{dx}x = 1\)

Step 3

Step 3 - Apply the variable rule.

In this problem: Since the expression is \(y\), its derivative is \(1\).

\(\frac{d}{dy}y = 1\)

Final Answer

\(1\)

Step 5

Step 5 - Combine and simplify.

In this problem: Combine the derivative from each separated term, then simplify the result.

\(f'\left(y\right) = 0 + 1 = 1\)

Step 1

Step 1 - Set up the terms in the sum.

In this problem: We are given a sum. The separate terms are \(x^{2}\), \(y^{2}\).

\(f\left(x\right) = x^{2} + y^{2}\)

Step 2

Step 2 - Write the sum rule formula.

In this problem: The sum rule says the derivative of a sum is the sum of the derivatives.

\(\frac{d}{dx}\left(f\left(x\right) + g\left(x\right)\right) = \frac{d}{dx}f\left(x\right) + \frac{d}{dx}g\left(x\right)\)

Step 3

Step 3 - Split the derivative across the addition signs.

In this problem: Apply the derivative to each separated term.

\(\frac{d}{dx}\left(x^{2} + y^{2}\right) = \frac{d}{dx}\left(x^{2}\right) + \frac{d}{dx}\left(y^{2}\right)\)

Step 4a

Step 4a - Differentiate the separated term \(x^{2}\).

In this problem: The separated term is \(x^{2}\). Use the power rule to get its derivative.

\(\frac{d}{dx}\left(x^{2}\right) = 2 x\)

Problem

\(f\left(x\right) = {x}^{2}\)

Approach

Step-by-step method

Set up the problem.
If a constant coefficient is present, apply the constant multiple rule first.
Write the power rule formula.
Substitute the exponent into the power rule.
Solve and simplify.

Step 1

Step 1 - Set up the problem.

In this problem: We are given \(x^{2}\). This is a power of \(x\), so the exponent is \(n = 2\).

\(f\left(x\right) = {x}^{2}\)

Step 2

Step 2 - Write the power rule formula.

In this problem: The power rule says to move the exponent to the front, then subtract 1 from the exponent.

\(\frac{d}{dx}{x}^{n} = n\,{x}^{n - 1}\)

Step 3

Step 3 - Substitute the given power into the formula.

In this problem: Here, \(n = 2\). Substitute this exponent into the power rule, but do not simplify yet.

\(\frac{d}{dx}{x}^{2} = 2{x}^{2 - 1}\)

Step 4

Step 4 - Solve and simplify.

In this problem: Now subtract 1 from the exponent and keep the result in power form. This gives \(2{x}^{1}\).

\(2{x}^{2 - 1} = 2{x}^{1} = 2 x\)

Final Answer

\(2 x\)

Step 4b

Step 4b - Differentiate the separated term \(y^{2}\).

In this problem: The separated term is \(y^{2}\). Use the constant rule to get its derivative.

\(\frac{d}{dx}\left(y^{2}\right) = 0\)

Problem

\(f\left(x\right) = y^{2}\)

Approach

Step-by-step method

Set up the problem.
Write the constant rule formula.
Apply the constant rule.

Step 1

Step 1 - Set up the problem.

In this problem: We are given \(y^{2}\). This expression does not contain \(x\), so it is a constant.

\(f\left(x\right) = y^{2}\)

Step 2

Step 2 - Write the constant rule formula.

In this problem: The constant rule says the derivative of any constant is \(0\).

\(\frac{d}{dx}c = 0\)

Step 3

Step 3 - Apply the constant rule.

In this problem: Since \(y^{2}\) is constant, its derivative is \(0\).

\(\frac{d}{dx}\left(y^{2}\right) = 0\)

Final Answer

\(0\)

Step 5

Step 5 - Combine and simplify.

In this problem: Combine the derivative from each separated term, then simplify the result.

\(f'\left(x\right) = 2 x + 0 = 2 x\)

Step 1

Step 1 - Set up the terms in the sum.

In this problem: We are given a sum. The separate terms are \(x^{2}\), \(y^{2}\).

\(f\left(y\right) = x^{2} + y^{2}\)

Step 2

Step 2 - Write the sum rule formula.

In this problem: The sum rule says the derivative of a sum is the sum of the derivatives.

\(\frac{d}{dy}\left(f\left(y\right) + g\left(y\right)\right) = \frac{d}{dy}f\left(y\right) + \frac{d}{dy}g\left(y\right)\)

Step 3

Step 3 - Split the derivative across the addition signs.

In this problem: Apply the derivative to each separated term.

\(\frac{d}{dy}\left(x^{2} + y^{2}\right) = \frac{d}{dy}\left(x^{2}\right) + \frac{d}{dy}\left(y^{2}\right)\)

Step 4a

Step 4a - Differentiate the separated term \(x^{2}\).

In this problem: The separated term is \(x^{2}\). Use the constant rule to get its derivative.

\(\frac{d}{dy}\left(x^{2}\right) = 0\)

Problem

\(f\left(y\right) = x^{2}\)

Approach

Step-by-step method

Set up the problem.
Write the constant rule formula.
Apply the constant rule.

Step 1

Step 1 - Set up the problem.

In this problem: We are given \(x^{2}\). This expression does not contain \(y\), so it is a constant.

\(f\left(y\right) = x^{2}\)

Step 2

Step 2 - Write the constant rule formula.

In this problem: The constant rule says the derivative of any constant is \(0\).

\(\frac{d}{dx}c = 0\)

Step 3

Step 3 - Apply the constant rule.

In this problem: Since \(x^{2}\) is constant, its derivative is \(0\).

\(\frac{d}{dy}\left(x^{2}\right) = 0\)

Final Answer

\(0\)

Step 4b

Step 4b - Differentiate the separated term \(y^{2}\).

In this problem: The separated term is \(y^{2}\). Use the power rule to get its derivative.

\(\frac{d}{dy}\left(y^{2}\right) = 2 y\)

Problem

\(f\left(y\right) = {y}^{2}\)

Approach

Step-by-step method

Set up the problem.
If a constant coefficient is present, apply the constant multiple rule first.
Write the power rule formula.
Substitute the exponent into the power rule.
Solve and simplify.

Step 1

Step 1 - Set up the problem.

In this problem: We are given \(y^{2}\). This is a power of \(x\), so the exponent is \(n = 2\).

\(f\left(y\right) = {y}^{2}\)

Step 2

Step 2 - Write the power rule formula.

In this problem: The power rule says to move the exponent to the front, then subtract 1 from the exponent.

\(\frac{d}{dy}{y}^{n} = n\,{y}^{n - 1}\)

Step 3

Step 3 - Substitute the given power into the formula.

In this problem: Here, \(n = 2\). Substitute this exponent into the power rule, but do not simplify yet.

\(\frac{d}{dy}{y}^{2} = 2{y}^{2 - 1}\)

Step 4

Step 4 - Solve and simplify.

In this problem: Now subtract 1 from the exponent and keep the result in power form. This gives \(2{y}^{1}\).

\(2{y}^{2 - 1} = 2{y}^{1} = 2 y\)

Final Answer

\(2 y\)

Step 5

Step 5 - Combine and simplify.

In this problem: Combine the derivative from each separated term, then simplify the result.

\(f'\left(y\right) = 0 + 2 y = 2 y\)

Step 3

Step 3 - Write the Lagrange system grad f = lambda grad g together with the constraint.

In this problem:

\(\begin{cases} 1 = \lambda\left(2 x\right) \\ 1 = \lambda\left(2 y\right) \\ x^{2} + y^{2} = 2 \end{cases}\)

Step 4

Step 4 - Solve the system for x, y, and lambda.

In this problem:

\(\begin{gathered} f(-1,-1) = -2 \\ f(1,1) = 2 \end{gathered}\)

Step 5

Step 5 - Evaluate f at each solution and pick the largest and smallest.

In this problem:

\(f_{\max} = 2,\quad f_{\min} = -2\)

Final Answer

\(f_{\max} = 2,\quad f_{\min} = -2\)

See Example 3 Hide Example 3

Problem

\(f = x + 2 y,\; g = x^{2} + y^{2} = 5\)

Approach

Step-by-step method

Set up the problem.
Find the gradients of f and g.
Write the Lagrange system grad f = lambda grad g together with the constraint.
Solve the system for x, y, and lambda.
Evaluate f at each solution and pick the largest and smallest.

Step 1

Step 1 - Set up the problem.

In this problem:

\(f = x + 2 y,\; g = x^{2} + y^{2} = 5\)

Step 2

Step 2 - Find the gradients of f and g.

In this problem:

\(\nabla f = \langle 1,2 \rangle,\quad \nabla g = \langle 2 x,2 y \rangle\)

Step 1

Step 1 - Set up the terms in the sum.

In this problem: We are given a sum. The separate terms are \(x\), \(2 y\).

\(f\left(x\right) = x + 2 y\)

Step 2

Step 2 - Write the sum rule formula.

In this problem: The sum rule says the derivative of a sum is the sum of the derivatives.

\(\frac{d}{dx}\left(f\left(x\right) + g\left(x\right)\right) = \frac{d}{dx}f\left(x\right) + \frac{d}{dx}g\left(x\right)\)

Step 3

Step 3 - Split the derivative across the addition signs.

In this problem: Apply the derivative to each separated term.

\(\frac{d}{dx}\left(x + 2 y\right) = \frac{d}{dx}\left(x\right) + \frac{d}{dx}\left(2 y\right)\)

Step 4a

Step 4a - Differentiate the separated term \(x\).

In this problem: The separated term is \(x\). Use the variable rule to get its derivative.

\(\frac{d}{dx}\left(x\right) = 1\)

Problem

\(f\left(x\right) = x\)

Approach

Step-by-step method

Set up the problem.
Write the variable rule formula.
Apply the variable rule.

Step 1

Step 1 - Set up the problem.

In this problem: We are given \(x\). This is just the variable \(x\).

\(f\left(x\right) = x\)

Step 2

Step 2 - Write the variable rule formula.

In this problem: The variable rule says the derivative of \(x\) with respect to \(x\) is \(1\).

\(\frac{d}{dx}x = 1\)

Step 3

Step 3 - Apply the variable rule.

In this problem: Since the expression is \(x\), its derivative is \(1\).

\(\frac{d}{dx}x = 1\)

Final Answer

\(1\)

Step 4b

Step 4b - Differentiate the separated term \(2 y\).

In this problem: The separated term is \(2 y\). Use the constant rule to get its derivative.

\(\frac{d}{dx}\left(2 y\right) = 0\)

Problem

\(f\left(x\right) = 2 y\)

Approach

Step-by-step method

Set up the problem.
Write the constant rule formula.
Apply the constant rule.

Step 1

Step 1 - Set up the problem.

In this problem: We are given \(2 y\). This expression does not contain \(x\), so it is a constant.

\(f\left(x\right) = 2 y\)

Step 2

Step 2 - Write the constant rule formula.

In this problem: The constant rule says the derivative of any constant is \(0\).

\(\frac{d}{dx}c = 0\)

Step 3

Step 3 - Apply the constant rule.

In this problem: Since \(2 y\) is constant, its derivative is \(0\).

\(\frac{d}{dx}\left(2 y\right) = 0\)

Final Answer

\(0\)

Step 5

Step 5 - Combine and simplify.

In this problem: Combine the derivative from each separated term, then simplify the result.

\(f'\left(x\right) = 1 + 0 = 1\)

Step 1

Step 1 - Set up the terms in the sum.

In this problem: We are given a sum. The separate terms are \(x\), \(2 y\).

\(f\left(y\right) = x + 2 y\)

Step 2

Step 2 - Write the sum rule formula.

In this problem: The sum rule says the derivative of a sum is the sum of the derivatives.

\(\frac{d}{dy}\left(f\left(y\right) + g\left(y\right)\right) = \frac{d}{dy}f\left(y\right) + \frac{d}{dy}g\left(y\right)\)

Step 3

Step 3 - Split the derivative across the addition signs.

In this problem: Apply the derivative to each separated term.

\(\frac{d}{dy}\left(x + 2 y\right) = \frac{d}{dy}\left(x\right) + \frac{d}{dy}\left(2 y\right)\)

Step 4a

Step 4a - Differentiate the separated term \(x\).

In this problem: The separated term is \(x\). Use the constant rule to get its derivative.

\(\frac{d}{dy}\left(x\right) = 0\)

Problem

\(f\left(y\right) = x\)

Approach

Step-by-step method

Set up the problem.
Write the constant rule formula.
Apply the constant rule.

Step 1

Step 1 - Set up the problem.

In this problem: We are given \(x\). This expression does not contain \(y\), so it is a constant.

\(f\left(y\right) = x\)

Step 2

Step 2 - Write the constant rule formula.

In this problem: The constant rule says the derivative of any constant is \(0\).

\(\frac{d}{dx}c = 0\)

Step 3

Step 3 - Apply the constant rule.

In this problem: Since \(x\) is constant, its derivative is \(0\).

\(\frac{d}{dy}\left(x\right) = 0\)

Final Answer

\(0\)

Step 4b

Step 4b - Differentiate the separated term \(2 y\).

In this problem: The separated term is \(2 y\). Use the power rule to get its derivative.

\(\frac{d}{dy}\left(2 y\right) = 2\)

Problem

\(f\left(y\right) = 2 y\)

Approach

Step-by-step method

Set up the problem.
If a constant coefficient is present, apply the constant multiple rule first.
Write the power rule formula.
Substitute the exponent into the power rule.
Solve and simplify.

Step 1

Step 1 - Set up the problem.

In this problem: We are given \(2\) times \(x\). Rewrite the variable part as \(x^{1}\), so the power rule can be used with exponent \(n = 1\).

\(f\left(y\right) = 2y \Leftrightarrow 2{y}^{1}\)

Step 2

Step 2 - Apply the constant multiple rule first.

In this problem: Before using the power rule, apply the constant multiple rule because \(2\) is a constant coefficient.

\(\frac{d}{dy}2{y}^{1} = 2\cdot \frac{d}{dy}{y}^{1}\)

Problem

\(f\left(y\right) = 2 y^{1}\)

Approach

Step-by-step method

Set up the problem.
Write the constant multiple rule formula.
Apply the constant multiple rule.

Step 1

Step 1 - Set up the coefficient and the variable part.

In this problem: We are given \(2 y^{1}\). The constant coefficient is \(2\), and the variable part is \(y^{1}\).

\(2 y^{1} = 2\cdot \left(y^{1}\right)\)

Step 2

Step 2 - Write the constant multiple rule formula.

In this problem: The constant multiple rule says a constant coefficient stays outside the derivative.

\(\frac{d}{dy}\left(c\cdot f\left(y\right)\right) = c\cdot \frac{d}{dy}f\left(y\right)\)

Step 3

Step 3 - Apply the constant multiple rule.

In this problem: Move the constant coefficient \(2\) outside the derivative and leave \(y^{1}\) inside the derivative.

\(\frac{d}{dy}\left(2 y^{1}\right) = 2\cdot \frac{d}{dy}\left(y^{1}\right)\)

Final Answer

\(2\cdot \frac{d}{dy}\left(y^{1}\right)\)

Step 3

Step 3 - Write the power rule formula.

In this problem: Now use the power rule on the remaining power of \(x\).

\(\frac{d}{dy}{y}^{n} = n\,{y}^{n - 1}\)

Step 4

Step 4 - Substitute the exponent into the formula.

In this problem: Use the rewritten power form from Step 1. Here, \(n = 1\). Substitute this exponent into the power rule, then keep the coefficient \(2\) outside.

\(2\cdot \frac{d}{dy}{y}^{1} = 2\cdot {y}^{1 - 1}\)

Step 5

Step 5 - Solve and simplify.

In this problem: Apply the power rule to \(y\), multiply by the constant \(2\), and keep the solving line in power form. This gives \(2\).

\(2\cdot {y}^{0} = 2\cdot 1 = 2\)

Final Answer

\(2\)

Step 5

Step 5 - Combine and simplify.

In this problem: Combine the derivative from each separated term, then simplify the result.

\(f'\left(y\right) = 0 + 2 = 2\)

Step 1

Step 1 - Set up the terms in the sum.

In this problem: We are given a sum. The separate terms are \(x^{2}\), \(y^{2}\).

\(f\left(x\right) = x^{2} + y^{2}\)

Step 2

Step 2 - Write the sum rule formula.

In this problem: The sum rule says the derivative of a sum is the sum of the derivatives.

\(\frac{d}{dx}\left(f\left(x\right) + g\left(x\right)\right) = \frac{d}{dx}f\left(x\right) + \frac{d}{dx}g\left(x\right)\)

Step 3

Step 3 - Split the derivative across the addition signs.

In this problem: Apply the derivative to each separated term.

\(\frac{d}{dx}\left(x^{2} + y^{2}\right) = \frac{d}{dx}\left(x^{2}\right) + \frac{d}{dx}\left(y^{2}\right)\)

Step 4a

Step 4a - Differentiate the separated term \(x^{2}\).

In this problem: The separated term is \(x^{2}\). Use the power rule to get its derivative.

\(\frac{d}{dx}\left(x^{2}\right) = 2 x\)

Problem

\(f\left(x\right) = {x}^{2}\)

Approach

Step-by-step method

Set up the problem.
If a constant coefficient is present, apply the constant multiple rule first.
Write the power rule formula.
Substitute the exponent into the power rule.
Solve and simplify.

Step 1

Step 1 - Set up the problem.

In this problem: We are given \(x^{2}\). This is a power of \(x\), so the exponent is \(n = 2\).

\(f\left(x\right) = {x}^{2}\)

Step 2

Step 2 - Write the power rule formula.

In this problem: The power rule says to move the exponent to the front, then subtract 1 from the exponent.

\(\frac{d}{dx}{x}^{n} = n\,{x}^{n - 1}\)

Step 3

Step 3 - Substitute the given power into the formula.

In this problem: Here, \(n = 2\). Substitute this exponent into the power rule, but do not simplify yet.

\(\frac{d}{dx}{x}^{2} = 2{x}^{2 - 1}\)

Step 4

Step 4 - Solve and simplify.

In this problem: Now subtract 1 from the exponent and keep the result in power form. This gives \(2{x}^{1}\).

\(2{x}^{2 - 1} = 2{x}^{1} = 2 x\)

Final Answer

\(2 x\)

Step 4b

Step 4b - Differentiate the separated term \(y^{2}\).

In this problem: The separated term is \(y^{2}\). Use the constant rule to get its derivative.

\(\frac{d}{dx}\left(y^{2}\right) = 0\)

Problem

\(f\left(x\right) = y^{2}\)

Approach

Step-by-step method

Set up the problem.
Write the constant rule formula.
Apply the constant rule.

Step 1

Step 1 - Set up the problem.

In this problem: We are given \(y^{2}\). This expression does not contain \(x\), so it is a constant.

\(f\left(x\right) = y^{2}\)

Step 2

Step 2 - Write the constant rule formula.

In this problem: The constant rule says the derivative of any constant is \(0\).

\(\frac{d}{dx}c = 0\)

Step 3

Step 3 - Apply the constant rule.

In this problem: Since \(y^{2}\) is constant, its derivative is \(0\).

\(\frac{d}{dx}\left(y^{2}\right) = 0\)

Final Answer

\(0\)

Step 5

Step 5 - Combine and simplify.

In this problem: Combine the derivative from each separated term, then simplify the result.

\(f'\left(x\right) = 2 x + 0 = 2 x\)

Step 1

Step 1 - Set up the terms in the sum.

In this problem: We are given a sum. The separate terms are \(x^{2}\), \(y^{2}\).

\(f\left(y\right) = x^{2} + y^{2}\)

Step 2

Step 2 - Write the sum rule formula.

In this problem: The sum rule says the derivative of a sum is the sum of the derivatives.

\(\frac{d}{dy}\left(f\left(y\right) + g\left(y\right)\right) = \frac{d}{dy}f\left(y\right) + \frac{d}{dy}g\left(y\right)\)

Step 3

Step 3 - Split the derivative across the addition signs.

In this problem: Apply the derivative to each separated term.

\(\frac{d}{dy}\left(x^{2} + y^{2}\right) = \frac{d}{dy}\left(x^{2}\right) + \frac{d}{dy}\left(y^{2}\right)\)

Step 4a

Step 4a - Differentiate the separated term \(x^{2}\).

In this problem: The separated term is \(x^{2}\). Use the constant rule to get its derivative.

\(\frac{d}{dy}\left(x^{2}\right) = 0\)

Problem

\(f\left(y\right) = x^{2}\)

Approach

Step-by-step method

Set up the problem.
Write the constant rule formula.
Apply the constant rule.

Step 1

Step 1 - Set up the problem.

In this problem: We are given \(x^{2}\). This expression does not contain \(y\), so it is a constant.

\(f\left(y\right) = x^{2}\)

Step 2

Step 2 - Write the constant rule formula.

In this problem: The constant rule says the derivative of any constant is \(0\).

\(\frac{d}{dx}c = 0\)

Step 3

Step 3 - Apply the constant rule.

In this problem: Since \(x^{2}\) is constant, its derivative is \(0\).

\(\frac{d}{dy}\left(x^{2}\right) = 0\)

Final Answer

\(0\)

Step 4b

Step 4b - Differentiate the separated term \(y^{2}\).

In this problem: The separated term is \(y^{2}\). Use the power rule to get its derivative.

\(\frac{d}{dy}\left(y^{2}\right) = 2 y\)

Problem

\(f\left(y\right) = {y}^{2}\)

Approach

Step-by-step method

Set up the problem.
If a constant coefficient is present, apply the constant multiple rule first.
Write the power rule formula.
Substitute the exponent into the power rule.
Solve and simplify.

Step 1

Step 1 - Set up the problem.

In this problem: We are given \(y^{2}\). This is a power of \(x\), so the exponent is \(n = 2\).

\(f\left(y\right) = {y}^{2}\)

Step 2

Step 2 - Write the power rule formula.

In this problem: The power rule says to move the exponent to the front, then subtract 1 from the exponent.

\(\frac{d}{dy}{y}^{n} = n\,{y}^{n - 1}\)

Step 3

Step 3 - Substitute the given power into the formula.

In this problem: Here, \(n = 2\). Substitute this exponent into the power rule, but do not simplify yet.

\(\frac{d}{dy}{y}^{2} = 2{y}^{2 - 1}\)

Step 4

Step 4 - Solve and simplify.

In this problem: Now subtract 1 from the exponent and keep the result in power form. This gives \(2{y}^{1}\).

\(2{y}^{2 - 1} = 2{y}^{1} = 2 y\)

Final Answer

\(2 y\)

Step 5

Step 5 - Combine and simplify.

In this problem: Combine the derivative from each separated term, then simplify the result.

\(f'\left(y\right) = 0 + 2 y = 2 y\)

Step 3

Step 3 - Write the Lagrange system grad f = lambda grad g together with the constraint.

In this problem:

\(\begin{cases} 1 = \lambda\left(2 x\right) \\ 2 = \lambda\left(2 y\right) \\ x^{2} + y^{2} = 5 \end{cases}\)

Step 4

Step 4 - Solve the system for x, y, and lambda.

In this problem:

\(\begin{gathered} f(-1,-2) = -5 \\ f(1,2) = 5 \end{gathered}\)

Step 5

Step 5 - Evaluate f at each solution and pick the largest and smallest.

In this problem:

\(f_{\max} = 5,\quad f_{\min} = -5\)

Final Answer

\(f_{\max} = 5,\quad f_{\min} = -5\)

See Example 4 Hide Example 4

Problem

\(f = x y,\; g = x + y = 10\)

Approach

Step-by-step method

Set up the problem.
Find the gradients of f and g.
Write the Lagrange system grad f = lambda grad g together with the constraint.
Solve the system for x, y, and lambda.
Evaluate f at each solution and pick the largest and smallest.

Step 1

Step 1 - Set up the problem.

In this problem:

\(f = x y,\; g = x + y = 10\)

Step 2

Step 2 - Find the gradients of f and g.

In this problem:

\(\nabla f = \langle y,x \rangle,\quad \nabla g = \langle 1,1 \rangle\)

Step 1

Step 1 - Set up the coefficient and the variable part.

In this problem: We are given \(x y\). The constant coefficient is \(y\), and the variable part is \(x\).

\(x y = y\cdot \left(x\right)\)

Step 2

Step 2 - Write the constant multiple rule formula.

In this problem: The constant multiple rule says a constant coefficient stays outside the derivative.

\(\frac{d}{dx}\left(c\cdot f\left(x\right)\right) = c\cdot \frac{d}{dx}f\left(x\right)\)

Step 3

Step 3 - Apply the constant multiple rule.

In this problem: Move the constant coefficient \(y\) outside the derivative and leave \(x\) inside the derivative.

\(\frac{d}{dx}\left(x y\right) = y\cdot \frac{d}{dx}\left(x\right)\)

Step 1

Step 1 - Set up the coefficient and the variable part.

In this problem: We are given \(x y\). The constant coefficient is \(x\), and the variable part is \(y\).

\(x y = x\cdot \left(y\right)\)

Step 2

Step 2 - Write the constant multiple rule formula.

In this problem: The constant multiple rule says a constant coefficient stays outside the derivative.

\(\frac{d}{dy}\left(c\cdot f\left(y\right)\right) = c\cdot \frac{d}{dy}f\left(y\right)\)

Step 3

Step 3 - Apply the constant multiple rule.

In this problem: Move the constant coefficient \(x\) outside the derivative and leave \(y\) inside the derivative.

\(\frac{d}{dy}\left(x y\right) = x\cdot \frac{d}{dy}\left(y\right)\)

Step 1

Step 1 - Set up the terms in the sum.

In this problem: We are given a sum. The separate terms are \(x\), \(y\).

\(f\left(x\right) = x + y\)

Step 2

Step 2 - Write the sum rule formula.

In this problem: The sum rule says the derivative of a sum is the sum of the derivatives.

\(\frac{d}{dx}\left(f\left(x\right) + g\left(x\right)\right) = \frac{d}{dx}f\left(x\right) + \frac{d}{dx}g\left(x\right)\)

Step 3

Step 3 - Split the derivative across the addition signs.

In this problem: Apply the derivative to each separated term.

\(\frac{d}{dx}\left(x + y\right) = \frac{d}{dx}\left(x\right) + \frac{d}{dx}\left(y\right)\)

Step 4a

Step 4a - Differentiate the separated term \(x\).

In this problem: The separated term is \(x\). Use the variable rule to get its derivative.

\(\frac{d}{dx}\left(x\right) = 1\)

Problem

\(f\left(x\right) = x\)

Approach

Step-by-step method

Set up the problem.
Write the variable rule formula.
Apply the variable rule.

Step 1

Step 1 - Set up the problem.

In this problem: We are given \(x\). This is just the variable \(x\).

\(f\left(x\right) = x\)

Step 2

Step 2 - Write the variable rule formula.

In this problem: The variable rule says the derivative of \(x\) with respect to \(x\) is \(1\).

\(\frac{d}{dx}x = 1\)

Step 3

Step 3 - Apply the variable rule.

In this problem: Since the expression is \(x\), its derivative is \(1\).

\(\frac{d}{dx}x = 1\)

Final Answer

\(1\)

Step 4b

Step 4b - Differentiate the separated term \(y\).

In this problem: The separated term is \(y\). Use the constant rule to get its derivative.

\(\frac{d}{dx}\left(y\right) = 0\)

Problem

\(f\left(x\right) = y\)

Approach

Step-by-step method

Set up the problem.
Write the constant rule formula.
Apply the constant rule.

Step 1

Step 1 - Set up the problem.

In this problem: We are given \(y\). This expression does not contain \(x\), so it is a constant.

\(f\left(x\right) = y\)

Step 2

Step 2 - Write the constant rule formula.

In this problem: The constant rule says the derivative of any constant is \(0\).

\(\frac{d}{dx}c = 0\)

Step 3

Step 3 - Apply the constant rule.

In this problem: Since \(y\) is constant, its derivative is \(0\).

\(\frac{d}{dx}\left(y\right) = 0\)

Final Answer

\(0\)

Step 5

Step 5 - Combine and simplify.

In this problem: Combine the derivative from each separated term, then simplify the result.

\(f'\left(x\right) = 1 + 0 = 1\)

Step 1

Step 1 - Set up the terms in the sum.

In this problem: We are given a sum. The separate terms are \(x\), \(y\).

\(f\left(y\right) = x + y\)

Step 2

Step 2 - Write the sum rule formula.

In this problem: The sum rule says the derivative of a sum is the sum of the derivatives.

\(\frac{d}{dy}\left(f\left(y\right) + g\left(y\right)\right) = \frac{d}{dy}f\left(y\right) + \frac{d}{dy}g\left(y\right)\)

Step 3

Step 3 - Split the derivative across the addition signs.

In this problem: Apply the derivative to each separated term.

\(\frac{d}{dy}\left(x + y\right) = \frac{d}{dy}\left(x\right) + \frac{d}{dy}\left(y\right)\)

Step 4a

Step 4a - Differentiate the separated term \(x\).

In this problem: The separated term is \(x\). Use the constant rule to get its derivative.

\(\frac{d}{dy}\left(x\right) = 0\)

Problem

\(f\left(y\right) = x\)

Approach

Step-by-step method

Set up the problem.
Write the constant rule formula.
Apply the constant rule.

Step 1

Step 1 - Set up the problem.

In this problem: We are given \(x\). This expression does not contain \(y\), so it is a constant.

\(f\left(y\right) = x\)

Step 2

Step 2 - Write the constant rule formula.

In this problem: The constant rule says the derivative of any constant is \(0\).

\(\frac{d}{dx}c = 0\)

Step 3

Step 3 - Apply the constant rule.

In this problem: Since \(x\) is constant, its derivative is \(0\).

\(\frac{d}{dy}\left(x\right) = 0\)

Final Answer

\(0\)

Step 4b

Step 4b - Differentiate the separated term \(y\).

In this problem: The separated term is \(y\). Use the variable rule to get its derivative.

\(\frac{d}{dy}\left(y\right) = 1\)

Problem

\(f\left(y\right) = y\)

Approach

Step-by-step method

Set up the problem.
Write the variable rule formula.
Apply the variable rule.

Step 1

Step 1 - Set up the problem.

In this problem: We are given \(y\). This is just the variable \(y\).

\(f\left(y\right) = y\)

Step 2

Step 2 - Write the variable rule formula.

In this problem: The variable rule says the derivative of \(y\) with respect to \(y\) is \(1\).

\(\frac{d}{dx}x = 1\)

Step 3

Step 3 - Apply the variable rule.

In this problem: Since the expression is \(y\), its derivative is \(1\).

\(\frac{d}{dy}y = 1\)

Final Answer

\(1\)

Step 5

Step 5 - Combine and simplify.

In this problem: Combine the derivative from each separated term, then simplify the result.

\(f'\left(y\right) = 0 + 1 = 1\)

Step 3

Step 3 - Write the Lagrange system grad f = lambda grad g together with the constraint.

In this problem:

\(\begin{cases} y = \lambda\left(1\right) \\ x = \lambda\left(1\right) \\ x + y = 10 \end{cases}\)

Step 4

Step 4 - Solve the system for x, y, and lambda.

In this problem:

\(\begin{gathered} f(5,5) = 25 \end{gathered}\)

Step 5

Step 5 - Evaluate f at each solution and pick the largest and smallest.

In this problem:

\(f_{\max} = 25,\quad f_{\min} = 25\)

Final Answer

\(f_{\max} = 25,\quad f_{\min} = 25\)