Answer
$\frac{{\partial f}}{{\partial s}} = {t^3} - 2s{t^3} + 3{s^2}t + 4s{t^2} + 6{s^2}{t^2}$
$\frac{{\partial f}}{{\partial t}} = {s^3} + 4{s^3}t + 4{s^2}t + 3s{t^2} - 3{s^2}{t^2}$
Work Step by Step
We have $f\left( {x,y,z} \right) = {x^2}y + {y^2}z$, where $x=s+t$, $y=s t$ and $z=2s-t$.
The partial derivatives are
$\frac{{\partial f}}{{\partial x}} = 2xy$, ${\ \ \ }$ $\frac{{\partial f}}{{\partial y}} = {x^2} + 2yz$, ${\ \ \ }$ $\frac{{\partial f}}{{\partial z}} = {y^2}$
$\frac{{\partial x}}{{\partial s}} = 1$, ${\ \ \ }$ $\frac{{\partial y}}{{\partial s}} = t$, ${\ \ \ }$ $\frac{{\partial z}}{{\partial s}} = 2$
$\frac{{\partial x}}{{\partial t}} = 1$, ${\ \ \ }$ $\frac{{\partial y}}{{\partial t}} = s$, ${\ \ \ }$ $\frac{{\partial z}}{{\partial t}} = - 1$
Using the Chain Rule, Eq. (2) and Eq. (3) of Section 15.6, we have
$\frac{{\partial f}}{{\partial s}} = \frac{{\partial f}}{{\partial x}}\frac{{\partial x}}{{\partial s}} + \frac{{\partial f}}{{\partial y}}\frac{{\partial y}}{{\partial s}} + \frac{{\partial f}}{{\partial z}}\frac{{\partial z}}{{\partial s}}$
$\frac{{\partial f}}{{\partial t}} = \frac{{\partial f}}{{\partial x}}\frac{{\partial x}}{{\partial t}} + \frac{{\partial f}}{{\partial y}}\frac{{\partial y}}{{\partial t}} + \frac{{\partial f}}{{\partial z}}\frac{{\partial z}}{{\partial t}}$
So,
(1) ${\ \ \ \ }$ $\frac{{\partial f}}{{\partial s}} = 2xy + \left( {{x^2} + 2yz} \right)t + 2{y^2}$
(2) ${\ \ \ \ }$ $\frac{{\partial f}}{{\partial t}} = 2xy + \left( {{x^2} + 2yz} \right)s - {y^2}$
Substituting $x=s+t$, $y=s t$ and $z=2s-t$ in equation (1) and (2) we obtain $\frac{{\partial f}}{{\partial s}}$ and $\frac{{\partial f}}{{\partial t}}$ in terms of $s$ and $t$:
$\frac{{\partial f}}{{\partial s}} = 2st\left( {s + t} \right) + {\left( {s + t} \right)^2}t + 2s{t^2}\left( {2s - t} \right) + 2{s^2}{t^2}$
$ = {t^3} - 2s{t^3} + 3{s^2}t + 4s{t^2} + 6{s^2}{t^2}$
$\frac{{\partial f}}{{\partial t}} = 2st\left( {s + t} \right) + {\left( {s + t} \right)^2}s + 2{s^2}t\left( {2s - t} \right) - {s^2}{t^2}$
$ = {s^3} + 4{s^3}t + 4{s^2}t + 3s{t^2} - 3{s^2}{t^2}$
