Partial Derivatives of Multivariable Functions: Examples and Solutions

Published on July 2, 2026

Calculus

Learn how to calculate the first partial derivatives of functions with multiple variables, with step-by-step solutions for f(x,y)=3x^2y+2x^3y and f(x,y,z)=x^3z+4xyz+z^2.

Question

Find the 1st partial derivative of the following functions A)f(x,y)=3x²y+2x³y B)f(x,y)=x³z +4xyz +z²

Answer

To find the first partial derivatives of a function of multiple variables, we differentiate with respect to one variable while treating all other variables as constants.

Part A)

Since there are two variables (

and

), we find the partial derivative with respect to

and the partial derivative with respect to

1. Partial derivative with respect to ( or ):
Treat

as a constant.

2. Partial derivative with respect to ( or ):
Treat

as a constant.

Part B)

Since there are three variables (

), we find the partial derivatives with respect to each.

1. Partial derivative with respect to ( or ):
Treat

and

as constants.

2. Partial derivative with respect to ( or ):
Treat

and

as constants.

3. Partial derivative with respect to ( or ):
Treat

and

as constants.

Theorems

• Sum Rule for Partial Derivatives

The partial derivative of a sum of functions is the sum of their partial derivatives. If

, then

and

• Constant Multiple Rule for Partial Derivatives

The partial derivative of a constant times a function is the constant times the partial derivative of the function. If

, where

is a constant, then

and

• Power Rule for Partial Derivatives

The partial derivative of

with respect to

. The partial derivative of

with respect to

. If

, then

and

• Partial Derivative of a Constant

The partial derivative of any term that does not contain the variable of differentiation is zero. If

, where

is a constant, then

and

🚀 Try Free Math Help Now!🏠 Learn More Powerful Features

Partial Derivatives of Multivariable Functions: Examples and Solutions

Question

Answer

Part A)

Part B)

Theorems

Related math solutions

永久解锁网页版

Related math solutions