Using the chain rule to compute $\frac{\partial{z}}{\partial{s}}$ given $z=x^2y$, $x=sin(st)$, $y = t^2+s^2$

Question

Im familiar with the chain rule when it's as such: You have a function z=f(x,y), f is differentiable and x=g(t) and y=h(t).

when that's the case I know that we have $\frac{dz}{dt}=\frac{\partial{z}}{\partial{x}}\frac{dx}{dt}+\frac{\partial{z}}{\partial{y}}\frac{dy}{dt}$

Since $\frac{\partial{z}}{\partial{s}}$ is not a term in the chain rule, how could I use it to find this?

Answer 1

In your case, since $x=g(s,t)$ and $y=h(s,t)$ and $z=z(x,y)$ you can say that $$ z=z(s,t) $$ and use the chain rule to calculate this. You will have $$ \dfrac{\partial z}{\partial s} = \dfrac{\partial z}{\partial x}\dfrac{\partial x}{\partial s} + \dfrac{\partial z}{\partial y}\dfrac{\partial y}{\partial s} = 2xyt\cos(st) + 2x^2s. $$ and in this last equation you can substitute $x$ and $y$ for $t$ and $s$.