Critical Points Notes

Published on: September 29, 2026

This Critical Points Calculator finds where a surface levels off and decides what each of those places is, and shows the working line by line. Both partial derivatives are set to zero first, since a level point has no slope in either direction. The second derivative test then separates a peak from a dip from a saddle. Following the same pattern every time keeps finding the points and classifying them as separate moves.

Step-by-step method

Set up the problem.
Find the first partial derivatives.
Solve f_x = 0 and f_y = 0 for the critical points.
Form the second-derivative test D = f_xx f_yy - (f_xy)^2.
Classify each critical point.

Formula:

\(D = f_{xx}f_{yy} - \left(f_{xy}\right)^{2}\)

See Example 1 Hide Example 1

Problem

\(f = x^{3} - 3 x y + y^{3}\)

Approach

Step-by-step method

Set up the problem.
Find the first partial derivatives.
Solve f_x = 0 and f_y = 0 for the critical points.
Form the second-derivative test D = f_xx f_yy - (f_xy)^2.
Classify each critical point.

Step 1

Step 1 - Set up the problem.

In this problem:

\(f = x^{3} - 3 x y + y^{3}\)

Step 2

Step 2 - Find the first partial derivatives.

In this problem:

\(f_x = 3 x^{2} - 3 y,\quad f_y = - 3 x + 3 y^{2}\)

Step 1

Step 1 - Set up the terms in the difference.

In this problem: We are given a difference. The separated terms are \(x^{3}\), \(3 x y\), \(y^{3}\).

\(f\left(x\right) = x^{3} - 3 x y + y^{3}\)

Step 2

Step 2 - Write the difference rule formula.

In this problem: The difference rule says the derivative of a difference is the difference 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 subtraction signs.

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

\(\frac{d}{dx}\left(x^{3} - 3 x y + y^{3}\right) = \frac{d}{dx}\left(x^{3}\right) - \frac{d}{dx}\left(3 x y\right) + \frac{d}{dx}\left(y^{3}\right)\)

Step 4a

Step 4a - Differentiate the separated term \(x^{3}\).

In this problem: The separated term is \(x^{3}\). Use the power rule to get its derivative.

\(\frac{d}{dx}\left(x^{3}\right) = 3 x^{2}\)

Problem

\(f\left(x\right) = {x}^{3}\)

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 \(x^{3}\). This is a power of \(x\), so the exponent is \(n = 3\).

\(f\left(x\right) = {x}^{3}\)

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, \(n = 3\). Substitute this exponent into the power rule, but do not simplify yet.

\(\frac{d}{dx}{x}^{3} = 3{x}^{3 - 1}\)

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 \(3{x}^{2}\).

\(3{x}^{3 - 1} = 3{x}^{2}\)

Final Answer

\(3 x^{2}\)

Step 4b

Step 4b - Differentiate the separated term \(3 x y\).

In this problem: The separated term is \(3 x y\). Use the power rule to get its derivative.

\(\frac{d}{dx}\left(3 x y\right) = 3 y\)

Problem

\(f\left(x\right) = 3 x y\)

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 \(3 y\) times \(x\). Rewrite the variable part as \(x^{1}\), so the power rule can be used with exponent \(n = 1\).

\(f\left(x\right) = 3 yx \Leftrightarrow 3 y{x}^{1}\)

Step 2

Step 2 - Apply the constant multiple rule first.

In this problem: Before using the power rule, apply the constant multiple rule because \(3 y\) is a constant coefficient.

\(\frac{d}{dx}3 y{x}^{1} = 3 y\cdot \frac{d}{dx}{x}^{1}\)

Problem

\(f\left(x\right) = x^{1} \cdot 3 y\)

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 \(x^{1} \cdot 3 y\). The constant coefficient is \(3 y\), and the variable part is \(x^{1}\).

\(x^{1} \cdot 3 y = 3 y\cdot \left(x^{1}\right)\)

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.

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

Step 3

Step 3 - Apply the constant multiple rule.

In this problem: Move the constant coefficient \(3 y\) outside the derivative and leave \(x^{1}\) inside the derivative.

\(\frac{d}{dx}\left(x^{1} \cdot 3 y\right) = 3 y\cdot \frac{d}{dx}\left(x^{1}\right)\)

Final Answer

\(3 y\cdot \frac{d}{dx}\left(x^{1}\right)\)

Step 3

Step 3 - Write the power rule formula.

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

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

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 \(3 y\) outside.

\(3 y\cdot \frac{d}{dx}{x}^{1} = 3 y\cdot {x}^{1 - 1}\)

Step 5

Step 5 - Solve and simplify.

In this problem: Apply the power rule to \(x\), multiply by the constant \(3 y\), and keep the solving line in power form. This gives \(3 y\).

\(3 y\cdot {x}^{0} = 3 y\cdot 1 = 3 y\)

Final Answer

\(3 y\)

Step 4c

Step 4c - Differentiate the separated term \(y^{3}\).

In this problem: The separated term is \(y^{3}\). Use the constant rule to get its derivative.

\(\frac{d}{dx}\left(y^{3}\right) = 0\)

Problem

\(f\left(x\right) = y^{3}\)

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 \(y^{3}\). This expression does not contain \(x\), so it is a constant.

\(f\left(x\right) = y^{3}\)

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 \(y^{3}\) is constant, its derivative is \(0\).

\(\frac{d}{dx}\left(y^{3}\right) = 0\)

Final Answer

\(0\)

Step 5

Step 5 - Combine and simplify.

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

\(f'\left(x\right) = 3 x^{2} - 3 y + 0 = 3 x^{2} - 3 y\)

Step 1

Step 1 - Set up the terms in the difference.

In this problem: We are given a difference. The separated terms are \(x^{3}\), \(3 x y\), \(y^{3}\).

\(f\left(x\right) = x^{3} - 3 x y + y^{3}\)

Step 2

Step 2 - Write the difference rule formula.

In this problem: The difference rule says the derivative of a difference is the difference 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 subtraction signs.

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

\(\frac{d}{dy}\left(x^{3} - 3 x y + y^{3}\right) = \frac{d}{dy}\left(x^{3}\right) - \frac{d}{dy}\left(3 x y\right) + \frac{d}{dy}\left(y^{3}\right)\)

Step 4a

Step 4a - Differentiate the separated term \(x^{3}\).

In this problem: The separated term is \(x^{3}\). Use the constant rule to get its derivative.

\(\frac{d}{dy}\left(x^{3}\right) = 0\)

Problem

\(f\left(y\right) = x^{3}\)

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 \(x^{3}\). This expression does not contain \(y\), so it is a constant.

\(f\left(y\right) = x^{3}\)

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 \(x^{3}\) is constant, its derivative is \(0\).

\(\frac{d}{dy}\left(x^{3}\right) = 0\)

Final Answer

\(0\)

Step 4b

Step 4b - Differentiate the separated term \(3 x y\).

In this problem: The separated term is \(3 x y\). Use the power rule to get its derivative.

\(\frac{d}{dy}\left(3 x y\right) = 3 x\)

Problem

\(f\left(y\right) = 3 x y\)

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 \(3 x\) times \(x\). Rewrite the variable part as \(x^{1}\), so the power rule can be used with exponent \(n = 1\).

\(f\left(y\right) = 3 xy \Leftrightarrow 3 x{y}^{1}\)

Step 2

Step 2 - Apply the constant multiple rule first.

In this problem: Before using the power rule, apply the constant multiple rule because \(3 x\) is a constant coefficient.

\(\frac{d}{dy}3 x{y}^{1} = 3 x\cdot \frac{d}{dy}{y}^{1}\)

Problem

\(f\left(y\right) = 3 x y^{1}\)

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 \(3 x y^{1}\). The constant coefficient is \(3 x\), and the variable part is \(y^{1}\).

\(3 x y^{1} = 3 x\cdot \left(y^{1}\right)\)

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 \(3 x\) outside the derivative and leave \(y^{1}\) inside the derivative.

\(\frac{d}{dy}\left(3 x y^{1}\right) = 3 x\cdot \frac{d}{dy}\left(y^{1}\right)\)

Final Answer

\(3 x\cdot \frac{d}{dy}\left(y^{1}\right)\)

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 \(3 x\) outside.

\(3 x\cdot \frac{d}{dy}{y}^{1} = 3 x\cdot {y}^{1 - 1}\)

Step 5

Step 5 - Solve and simplify.

In this problem: Apply the power rule to \(y\), multiply by the constant \(3 x\), and keep the solving line in power form. This gives \(3 x\).

\(3 x\cdot {y}^{0} = 3 x\cdot 1 = 3 x\)

Final Answer

\(3 x\)

Step 4c

Step 4c - Differentiate the separated term \(y^{3}\).

In this problem: The separated term is \(y^{3}\). Use the power rule to get its derivative.

\(\frac{d}{dy}\left(y^{3}\right) = 3 y^{2}\)

Problem

\(f\left(y\right) = {y}^{3}\)

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 \(y^{3}\). This is a power of \(x\), so the exponent is \(n = 3\).

\(f\left(y\right) = {y}^{3}\)

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, \(n = 3\). Substitute this exponent into the power rule, but do not simplify yet.

\(\frac{d}{dy}{y}^{3} = 3{y}^{3 - 1}\)

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 \(3{y}^{2}\).

\(3{y}^{3 - 1} = 3{y}^{2}\)

Final Answer

\(3 y^{2}\)

Step 5

Step 5 - Combine and simplify.

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

\(f'\left(x\right) = 0 - 3 x + 3 y^{2} = - 3 x + 3 y^{2}\)

Step 3

Step 3 - Solve f_x = 0 and f_y = 0 for the critical points.

In this problem:

\(\text{Critical points: }(0, 0),\; (1, 1)\)

Step 4

Step 4 - Form the second-derivative test D = f_xx f_yy - (f_xy)^2.

In this problem:

\(f_{xx} = 6 x,\quad f_{yy} = 6 y,\quad f_{xy} = -3\)

Step 5

Step 5 - Classify each critical point.

In this problem:

\(\begin{gathered} (0, 0): \text{a saddle point} \\ (1, 1): \text{a local minimum} \end{gathered}\)

Final Answer

\(\begin{gathered} (0, 0): \text{a saddle point} \\ (1, 1): \text{a local minimum} \end{gathered}\)

See Example 2 Hide Example 2

Problem

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

Approach

Step-by-step method

Set up the problem.
Find the first partial derivatives.
Solve f_x = 0 and f_y = 0 for the critical points.
Form the second-derivative test D = f_xx f_yy - (f_xy)^2.
Classify each critical point.

Step 1

Step 1 - Set up the problem.

In this problem:

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

Step 2

Step 2 - Find the first partial derivatives.

In this problem:

\(f_x = 2 x,\quad f_y = 2 y\)

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 - Solve f_x = 0 and f_y = 0 for the critical points.

In this problem:

\(\text{Critical points: }(0, 0)\)

Step 4

Step 4 - Form the second-derivative test D = f_xx f_yy - (f_xy)^2.

In this problem:

\(f_{xx} = 2,\quad f_{yy} = 2,\quad f_{xy} = 0\)

Step 5

Step 5 - Classify each critical point.

In this problem:

\(\begin{gathered} (0, 0): \text{a local minimum} \end{gathered}\)

Final Answer

\(\begin{gathered} (0, 0): \text{a local minimum} \end{gathered}\)

See Example 3 Hide Example 3

Problem

\(f = x^{2} - 4 x + y^{2} + 2 y\)

Approach

Step-by-step method

Set up the problem.
Find the first partial derivatives.
Solve f_x = 0 and f_y = 0 for the critical points.
Form the second-derivative test D = f_xx f_yy - (f_xy)^2.
Classify each critical point.

Step 1

Step 1 - Set up the problem.

In this problem:

\(f = x^{2} - 4 x + y^{2} + 2 y\)

Step 2

Step 2 - Find the first partial derivatives.

In this problem:

\(f_x = 2 x - 4,\quad f_y = 2 y + 2\)

Step 1

Step 1 - Set up the terms in the difference.

In this problem: We are given a difference. The separated terms are \(x^{2}\), \(4 x\), \(y^{2}\), \(2 y\).

\(f\left(x\right) = x^{2} - 4 x + y^{2} + 2 y\)

Step 2

Step 2 - Write the difference rule formula.

In this problem: The difference rule says the derivative of a difference is the difference 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 subtraction signs.

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

\(\frac{d}{dx}\left(x^{2} - 4 x + y^{2} + 2 y\right) = \frac{d}{dx}\left(x^{2}\right) - \frac{d}{dx}\left(4 x\right) + \frac{d}{dx}\left(y^{2}\right) + \frac{d}{dx}\left(2 y\right)\)

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 \(4 x\).

In this problem: The separated term is \(4 x\). Use the power rule to get its derivative.

\(\frac{d}{dx}\left(4 x\right) = 4\)

Problem

\(f\left(x\right) = 4 x\)

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 \(4\) times \(x\). Rewrite the variable part as \(x^{1}\), so the power rule can be used with exponent \(n = 1\).

\(f\left(x\right) = 4x \Leftrightarrow 4{x}^{1}\)

Step 2

Step 2 - Apply the constant multiple rule first.

In this problem: Before using the power rule, apply the constant multiple rule because \(4\) is a constant coefficient.

\(\frac{d}{dx}4{x}^{1} = 4\cdot \frac{d}{dx}{x}^{1}\)

Problem

\(f\left(x\right) = 4 x^{1}\)

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 \(4 x^{1}\). The constant coefficient is \(4\), and the variable part is \(x^{1}\).

\(4 x^{1} = 4\cdot \left(x^{1}\right)\)

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.

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

Step 3

Step 3 - Apply the constant multiple rule.

In this problem: Move the constant coefficient \(4\) outside the derivative and leave \(x^{1}\) inside the derivative.

\(\frac{d}{dx}\left(4 x^{1}\right) = 4\cdot \frac{d}{dx}\left(x^{1}\right)\)

Final Answer

\(4\cdot \frac{d}{dx}\left(x^{1}\right)\)

Step 3

Step 3 - Write the power rule formula.

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

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

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 \(4\) outside.

\(4\cdot \frac{d}{dx}{x}^{1} = 4\cdot {x}^{1 - 1}\)

Step 5

Step 5 - Solve and simplify.

In this problem: Apply the power rule to \(x\), multiply by the constant \(4\), and keep the solving line in power form. This gives \(4\).

\(4\cdot {x}^{0} = 4\cdot 1 = 4\)

Final Answer

\(4\)

Step 4c

Step 4c - 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 4d

Step 4d - 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, keeping the signs, then simplify.

\(f'\left(x\right) = 2 x - 4 + 0 + 0 = 2 x - 4\)

Step 1

Step 1 - Set up the terms in the sum.

In this problem: We are given a sum. The separate terms are \(x \left(x - 4\right)\), \(y^{2}\), \(2 y\).

\(f\left(y\right) = x \left(x - 4\right) + y^{2} + 2 y\)

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.

\(\frac{d}{dy}\left(x \left(x - 4\right) + y^{2} + 2 y\right) = \frac{d}{dy}\left(x \left(x - 4\right)\right) + \frac{d}{dy}\left(y^{2}\right) + \frac{d}{dy}\left(2 y\right)\)

Step 4a

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

In this problem: The separated term is \(x \left(x - 4\right)\). Use the constant rule to get its derivative.

\(\frac{d}{dy}\left(x \left(x - 4\right)\right) = 0\)

Problem

\(f\left(y\right) = x^{2} - 4 x\)

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 \(x^{2} - 4 x\). This expression does not contain \(y\), so it is a constant.

\(f\left(y\right) = x^{2} - 4 x\)

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 \(x^{2} - 4 x\) is constant, its derivative is \(0\).

\(\frac{d}{dy}\left(x^{2} - 4 x\right) = 0\)

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 4c

Step 4c - 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.

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

Step 3

Step 3 - Solve f_x = 0 and f_y = 0 for the critical points.

In this problem:

\(\text{Critical points: }(2, -1)\)

Step 4

Step 4 - Form the second-derivative test D = f_xx f_yy - (f_xy)^2.

In this problem:

\(f_{xx} = 2,\quad f_{yy} = 2,\quad f_{xy} = 0\)

Step 5

Step 5 - Classify each critical point.

In this problem:

\(\begin{gathered} (2, -1): \text{a local minimum} \end{gathered}\)

Final Answer

\(\begin{gathered} (2, -1): \text{a local minimum} \end{gathered}\)

See Example 4 Hide Example 4

Problem

\(f = x^{2} - y^{2}\)

Approach

Step-by-step method

Set up the problem.
Find the first partial derivatives.
Solve f_x = 0 and f_y = 0 for the critical points.
Form the second-derivative test D = f_xx f_yy - (f_xy)^2.
Classify each critical point.

Step 1

Step 1 - Set up the problem.

In this problem:

\(f = x^{2} - y^{2}\)

Step 2

Step 2 - Find the first partial derivatives.

In this problem:

\(f_x = 2 x,\quad f_y = - 2 y\)

Step 1

Step 1 - Set up the terms in the difference.

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

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

Step 2

Step 2 - Write the difference rule formula.

In this problem: The difference rule says the derivative of a difference is the difference 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 subtraction signs.

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

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

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, keeping the signs, then simplify.

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

Step 1

Step 1 - Set up the terms in the difference.

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

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

Step 2

Step 2 - Write the difference rule formula.

In this problem: The difference rule says the derivative of a difference is the difference 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 subtraction signs.

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

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

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, keeping the signs, then simplify.

\(f'\left(x\right) = 0 - 2 y = - 2 y\)

Step 3

Step 3 - Solve f_x = 0 and f_y = 0 for the critical points.

In this problem:

\(\text{Critical points: }(0, 0)\)

Step 4

Step 4 - Form the second-derivative test D = f_xx f_yy - (f_xy)^2.

In this problem:

\(f_{xx} = 2,\quad f_{yy} = -2,\quad f_{xy} = 0\)

Step 5

Step 5 - Classify each critical point.

In this problem:

\(\begin{gathered} (0, 0): \text{a saddle point} \end{gathered}\)

Final Answer

\(\begin{gathered} (0, 0): \text{a saddle point} \end{gathered}\)