Partial Derivatives Notes

Published on: September 29, 2026

This Partial Derivatives Calculator differentiates a function of several variables with respect to each one in turn, and shows the working line by line. While one variable is being differentiated the others are treated as constants, which is what makes each partial an ordinary derivative. Every variable in the function gets its own result. Following the same pattern every time keeps the held-constant variables from being differentiated by mistake.

Step-by-step method

Set up the problem.
For each variable, hold the other variables constant.
Differentiate with respect to that one variable using the ordinary derivative rules.
Write each partial derivative.

Formula:

\(\frac{\partial f}{\partial x} = f_{x}, \quad \frac{\partial f}{\partial y} = f_{y}\)

See Example 1 Hide Example 1

Problem

\(f = x^{3} + x y^{2}\)

Approach

Step-by-step method

Set up the problem.
For each variable, hold the other variables constant.
Differentiate with respect to that one variable using the ordinary derivative rules.
Write each partial derivative.

Step 1

Step 1 - Partial derivative with respect to x.

In this problem:

\(\frac{\partial f}{\partial x}\)

Step 2

Step 2 - Set up the terms in the sum.

In this problem:

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

Step 3

Step 3 - Write the sum rule formula.

In this problem:

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

Step 4

Step 4 - Split the derivative across the addition signs.

In this problem:

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

Step 5

Step 5 - Differentiate the separated term \(x^{3}\).

In this problem:

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

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 6

Step 6 - Differentiate the separated term \(x y^{2}\).

In this problem:

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

Problem

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

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 \(x y^{2}\). This is a constant multiple of a power of \(x\). The coefficient is \(y^{2}\), the power part is \(x\), and the exponent is \(n = 1\).

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

Step 2

Step 2 - Apply the constant multiple rule first.

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

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

Problem

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

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 \(x^{1} y^{2}\). The constant coefficient is \(y^{2}\), and the variable part is \(x^{1}\).

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

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 \(y^{2}\) outside the derivative and leave \(x^{1}\) inside the derivative.

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

Final Answer

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

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: Here, \(n = 1\). Substitute this exponent into the power rule, then keep the coefficient \(y^{2}\) outside.

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

Step 5

Step 5 - Solve and simplify.

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

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

Final Answer

\(y^{2}\)

Step 7

Step 7 - Combine and simplify.

In this problem:

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

Step 8

Step 8 - Partial derivative with respect to y.

In this problem:

\(\frac{\partial f}{\partial y}\)

Step 9

Step 9 - Set up the terms in the sum.

In this problem:

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

Step 10

Step 10 - Write the sum rule formula.

In this problem:

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

Step 11

Step 11 - Split the derivative across the addition signs.

In this problem:

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

Step 12

Step 12 - Differentiate the separated term \(x^{3}\).

In this problem:

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

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 13

Step 13 - Differentiate the separated term \(x y^{2}\).

In this problem:

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

Problem

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

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 \(x y^{2}\). This is a constant multiple of a power of \(x\). The coefficient is \(x\), the power part is \(y^{2}\), and the exponent is \(n = 2\).

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

Step 2

Step 2 - Apply the constant multiple rule first.

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

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

Problem

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

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 \(x y^{2}\). The constant coefficient is \(x\), and the variable part is \(y^{2}\).

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

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 \(y^{2}\) inside the derivative.

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

Final Answer

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

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: Here, \(n = 2\). Substitute this exponent into the power rule, then keep the coefficient \(x\) outside.

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

Step 5

Step 5 - Solve and simplify.

In this problem: Apply the power rule to \(y^{2}\), multiply by the constant \(x\), and keep the solving line in power form. This gives \(2 x{y}^{1}\).

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

Final Answer

\(2 x y\)

Step 14

Step 14 - Combine and simplify.

In this problem:

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

Final Answer

\(\frac{\partial f}{\partial x} = 3 x^{2} + y^{2},\quad \frac{\partial f}{\partial y} = 2 x y\)

See Example 2 Hide Example 2

Problem

\(f = \sin{\left(x \right)} \cos{\left(y \right)}\)

Approach

Step-by-step method

Set up the problem.
For each variable, hold the other variables constant.
Differentiate with respect to that one variable using the ordinary derivative rules.
Write each partial derivative.

Step 1

Step 1 - Partial derivative with respect to x.

In this problem:

\(\frac{\partial f}{\partial x}\)

Step 2

Step 2 - Identify the trigonometric function and any constant multiple.

In this problem:

\(f\left(x\right) = \sin{\left(x \right)} \cos{\left(y \right)}\)

Step 3

Step 3 - Write the derivative rule for that trigonometric function.

In this problem:

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

Step 4

Step 4 - Apply the rule, keeping the constant multiple in front.

In this problem:

\(f'\left(x\right) = \cos{\left(y \right)} \cdot \left(\cos{\left(x \right)}\right) = \cos{\left(x \right)} \cos{\left(y \right)}\)

Step 5

Step 5 - Partial derivative with respect to y.

In this problem:

\(\frac{\partial f}{\partial y}\)

Step 6

Step 6 - Identify the trigonometric function and any constant multiple.

In this problem:

\(f\left(y\right) = \sin{\left(x \right)} \cos{\left(y \right)}\)

Step 7

Step 7 - Write the derivative rule for that trigonometric function.

In this problem:

\(\frac{d}{dy}\left(\cos{\left(y \right)}\right) = - \sin{\left(y \right)}\)

Step 8

Step 8 - Apply the rule, keeping the constant multiple in front.

In this problem:

\(f'\left(y\right) = \sin{\left(x \right)} \cdot \left(- \sin{\left(y \right)}\right) = - \sin{\left(x \right)} \sin{\left(y \right)}\)

Final Answer

\(\frac{\partial f}{\partial x} = \cos{\left(x \right)} \cos{\left(y \right)},\quad \frac{\partial f}{\partial y} = - \sin{\left(x \right)} \sin{\left(y \right)}\)

See Example 3 Hide Example 3

Problem

\(f = x^{2} y + 3 y^{3}\)

Approach

Step-by-step method

Set up the problem.
For each variable, hold the other variables constant.
Differentiate with respect to that one variable using the ordinary derivative rules.
Write each partial derivative.

Step 1

Step 1 - Partial derivative with respect to x.

In this problem:

\(\frac{\partial f}{\partial x}\)

Step 2

Step 2 - Set up the terms in the sum.

In this problem:

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

Step 3

Step 3 - Write the sum rule formula.

In this problem:

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

Step 4

Step 4 - Split the derivative across the addition signs.

In this problem:

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

Step 5

Step 5 - Differentiate the separated term \(x^{2} y\).

In this problem:

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

Problem

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

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 \(x^{2} y\). This is a constant multiple of a power of \(x\). The coefficient is \(y\), the power part is \(x^{2}\), and the exponent is \(n = 2\).

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

Step 2

Step 2 - Apply the constant multiple rule first.

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

\(\frac{d}{dx}y{x}^{2} = y\cdot \frac{d}{dx}{x}^{2}\)

Problem

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

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 \(x^{2} y\). The constant coefficient is \(y\), and the variable part is \(x^{2}\).

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

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 \(x^{2}\) inside the derivative.

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

Final Answer

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

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: Here, \(n = 2\). Substitute this exponent into the power rule, then keep the coefficient \(y\) outside.

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

Step 5

Step 5 - Solve and simplify.

In this problem: Apply the power rule to \(x^{2}\), multiply by the constant \(y\), and keep the solving line in power form. This gives \(2 y{x}^{1}\).

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

Final Answer

\(2 x y\)

Step 6

Step 6 - Differentiate the separated term \(3 y^{3}\).

In this problem:

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

Problem

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

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

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

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

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

Final Answer

\(0\)

Step 7

Step 7 - Combine and simplify.

In this problem:

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

Step 8

Step 8 - Partial derivative with respect to y.

In this problem:

\(\frac{\partial f}{\partial y}\)

Step 9

Step 9 - Set up the terms in the sum.

In this problem:

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

Step 10

Step 10 - Write the sum rule formula.

In this problem:

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

Step 11

Step 11 - Split the derivative across the addition signs.

In this problem:

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

Step 12

Step 12 - Differentiate the separated term \(x^{2} y\).

In this problem:

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

Problem

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

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 \(x^{2} y\). This is a constant multiple of a power of \(x\). The coefficient is \(x^{2}\), the power part is \(y\), and the exponent is \(n = 1\).

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

Step 2

Step 2 - Apply the constant multiple rule first.

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

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

Problem

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

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 \(x^{2} y^{1}\). The constant coefficient is \(x^{2}\), and the variable part is \(y^{1}\).

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

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 \(x^{2}\) outside the derivative and leave \(y^{1}\) inside the derivative.

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

Final Answer

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

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: Here, \(n = 1\). Substitute this exponent into the power rule, then keep the coefficient \(x^{2}\) outside.

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

Step 5

Step 5 - Solve and simplify.

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

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

Final Answer

\(x^{2}\)

Step 13

Step 13 - Differentiate the separated term \(3 y^{3}\).

In this problem:

\(\frac{d}{dy}\left(3 y^{3}\right) = 9 y^{2}\)

Problem

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

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 \(3 y^{3}\). This is a constant multiple of a power of \(x\). The coefficient is \(3\), the power part is \(y^{3}\), and the exponent is \(n = 3\).

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

Step 2

Step 2 - Apply the constant multiple rule first.

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

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

Problem

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

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

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

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

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

Final Answer

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

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: Here, \(n = 3\). Substitute this exponent into the power rule, then keep the coefficient \(3\) outside.

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

Step 5

Step 5 - Solve and simplify.

In this problem: Apply the power rule to \(y^{3}\), multiply by the constant \(3\), and keep the solving line in power form. This gives \(9{y}^{2}\).

\(3\cdot 3{y}^{2} = 9{y}^{2}\)

Final Answer

\(9 y^{2}\)

Step 14

Step 14 - Combine and simplify.

In this problem:

\(f'\left(y\right) = x^{2} + 9 y^{2}\)

Final Answer

\(\frac{\partial f}{\partial x} = 2 x y,\quad \frac{\partial f}{\partial y} = x^{2} + 9 y^{2}\)

See Example 4 Hide Example 4

Problem

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

Approach

Step-by-step method

Set up the problem.
For each variable, hold the other variables constant.
Differentiate with respect to that one variable using the ordinary derivative rules.
Write each partial derivative.

Step 1

Step 1 - Partial derivative with respect to x.

In this problem:

\(\frac{\partial f}{\partial x}\)

Step 2

Step 2 - Identify the exponential function and any constant multiple.

In this problem:

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

Step 3

Step 3 - Write the derivative rule for that exponential function.

In this problem:

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

Step 4

Step 4 - Apply the rule, keeping the constant multiple in front.

In this problem:

\(f'\left(x\right) = \sin{\left(y \right)} \cdot \left(e^{x}\right) = e^{x} \sin{\left(y \right)}\)

Step 5

Step 5 - Partial derivative with respect to y.

In this problem:

\(\frac{\partial f}{\partial y}\)

Step 6

Step 6 - Identify the trigonometric function and any constant multiple.

In this problem:

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

Step 7

Step 7 - Write the derivative rule for that trigonometric function.

In this problem:

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

Step 8

Step 8 - Apply the rule, keeping the constant multiple in front.

In this problem:

\(f'\left(y\right) = e^{x} \cdot \left(\cos{\left(y \right)}\right) = e^{x} \cos{\left(y \right)}\)

Final Answer

\(\frac{\partial f}{\partial x} = e^{x} \sin{\left(y \right)},\quad \frac{\partial f}{\partial y} = e^{x} \cos{\left(y \right)}\)