How to use multivariate chain rule and table of values to find partial derivatives?

Question

I don't know what the first thing I should do for this question is. How can the principles of the multivariate form of the chain rule be applied to solve this question? Thanks. I am trying using this version of the chain rule

Answer 1

The first thing is to understand how the chain rule applies to your problem. As the problem indicates you have $x=e^u + \sin v, y = e^u + \cos v.$ Where does that fit into the chain rule formula? (The unfortunate thing with this statement of the chain rule is that $u$ means different things on the two sides of the equation. On the left-hand side it means $g,$ and on the right-hand side, it means $f$. This isn't a peculiarity of the chain rule for several variables. The chain rule in one variable also exhibits this phenomenon.)

Anyway, on the left-hand side, $u$ means $g$ and $t_i$ is one of the variables we're differentiating with respect to: say $u$. Then on the right-hand side, $u$ means $f,$ and the $x_i$ are the variables in the definition of $f$. So we can take $x_1 = x, x_2 = y$. That gives us: $$ \frac{\partial g}{\partial u} = \frac{\partial f}{\partial x}\frac{\partial x}{\partial u}+\frac{\partial f}{\partial y}\frac{\partial y}{\partial u} $$ We get a completely similar equation for the derivative with resect to $v$.

I now you have to calculate the partial derivatives of $x$ and $y$ with respect to $u$ and $v$ and substitute into the formulas above.

Just in case, I'll mention that in the problem $g_u,$ for example, means $\frac{\partial g}{\partial u}.$

Try it now.