https://i.stack.imgur.com/hKv7e.png
Question 10
(a) Consider the function f given by:
$f(x, y)$ = $x^3$ + $3xy^2$ + $12y^2$ − $75x$.
(i) Calculate the first and second order partial derivatives $fx$, $fy,$ $fxx$, $fyy$, $fxy$, and $fyx$.
(ii) Find and classify all of the stationary points of $f$
(b) Suppose that $z = g(x, y)$, $x = s+t$, and $y = st$, where all first and second order partial derivatives of g exist and are continuous.
Show that $\frac{∂^2z}{∂s∂t}$ = $\frac{∂^2g}{∂x^2}$ + $x\frac{∂^2g}{∂x∂y}$ + $y\frac{∂^2g}{∂y^2}$ + $\frac{∂g}{∂y}$
Can someone please explain how to do question 10b, I know that $\frac{∂x}{∂s}$ = 1, $\frac{∂x}{∂t}$ = 1, $\frac{∂y}{∂s}$ = t, $\frac{∂y}{∂t}$ = s
but I have no idea what to do from here
\begin{align} z_t &= g_x \, x_t + g_y \, y_t = g_x + g_y \, s \\ z_{st} &= g_{xx} \, x_s + g_{yx} \, y_s + (g_{xy} \, x_s + g_{yy} \, y_s) s + g_y \\ &= g_{xx} + g_{xy} \, (t + s) + g_{yy}\,ts + g_y \\ &= g_{xx} + g_{xy}\, x + g_{yy} \, y + g_y \end{align}