Question

Find all first and second partial derivatives of the function f(x,y,z)=x2y3z4. There are six distinct second derivatives to compute.

Answer 1

The first partial derivatives of f(x,y,z)=x²y³z⁴ are 2xy³z⁴, 3x²y²z⁴, and 4x²y³z³. The second partial derivatives are 2y³z⁴, 6xy²z⁴, 12x²y³z², 6xy²z⁴, 8xy³z³, and 12x²y²z³.

Step-by-step explanation:

To find the first and second partial derivatives of the function f(x,y,z)=x2y3z4, we need to differentiate with respect to each variable. Here are the step-by-step calculations:

First partial derivatives:

∂f/∂x = 2xy³z⁴

∂f/∂y = 3x²y²z⁴

∂f/∂z = 4x²y³z³

Second partial derivatives:

∂²f/∂x² = 2y³z⁴

∂²f/∂y² = 6xy²z⁴

∂²f/∂z² = 12x²y³z²

∂²f/∂x∂y = 6xy²z⁴

∂²f/∂x∂z = 8xy³z³

∂²f/∂y∂z = 12x²y²z³