Chapter 14.6, Problem 1E

To determine

Derivatives of the given function.

$\begin{array}{l}f(x,y,z)=x^2{y}^3+z^4\\ x=s^2\\ y=s{t}^2\\ z=s^{2}t\end{array}$

(a) We have to calculate the primary derivatives,

$\frac{\partial f}{\partial x},\frac{\partial f}{\partial y},\frac{\partial f}{\partial z}$

(b) We have to calculate,

$\frac{\partial x}{\partial s},\frac{\partial y}{\partial s},\frac{\partial z}{\partial s}$

(c) We have to complete the derivatives $\left(\frac{\partial f}{\partial s}\right)$ by chain rule.

$\frac{\partial f}{\partial s}=\frac{\partial f}{\partial x}.\frac{\partial f}{\partial x}+\frac{\partial f}{\partial y}.\frac{\partial y}{\partial s}+\frac{\partial f}{\partial z}.\frac{\partial z}{\partial s}$

Answer should be in terms of independent variable s, t.