Suppose you are given that $f$ is a differentiable function of two variables $u$ and $v$. Let $g(r,s)=f(r^2-s^2, 2rs)$. Compute $\nabla g(r,s)$ in terms of $\partial f\over\partial u$, $\partial f\over\partial v$ and other functions.
I expressed $f$ as $f(u,v)=f(r^2-s^2, 2rs)$ and set $u=r^2-s^2$ and $v=2rs$. To calculate the gradient I did:
$$\nabla g(r,s)=\frac{\partial g}{\partial r}\mathbf i + \frac{\partial g}{\partial s}\mathbf j$$
From there:
$$=\left({\partial f\over\partial u}{\partial u\over\partial r}\right)\mathbf i + \left({\partial f\over\partial v}{\partial v\over\partial s}\right)\mathbf j$$
This is the step where I lost points but I can't seem to figure out where I went wrong.
In replacing $\frac{\partial g}{\partial r}$ with $\left({\partial f\over\partial u}{\partial u\over\partial r}\right)$, you are treating $f$ as if it were a function of $u$ alone, and similarly in replacing $\frac{\partial g}{\partial s}$ with $\left({\partial f\over\partial v}{\partial v\over\partial s}\right)$, you are treating $f$ as if it were a function of $v$ alone.
The correct formulas are:$$\frac{\partial g}{\partial r} = {\partial f\over\partial u}{\partial u\over\partial r} + {\partial f\over\partial v}{\partial v\over\partial r}\\\frac{\partial g}{\partial s} = {\partial f\over\partial u}{\partial u\over\partial s} + {\partial f\over\partial v}{\partial v\over\partial s}$$