Chain rule with multiple variables

Question

I am given $t(r, θ) = u(r \cos θ, r \sin θ)$, and I need to find $t_r$ and $t_θ$. I know that I need to apply the chain rule as $t$ is a composition of $u$ and another function, but how? I'm confused since $u$ takes two inputs. Do I differentiate separately for each of the two inputs?

Answer 1

$$ t_\theta = \frac{\partial u}{\partial\theta} = \frac{\partial u}{\partial x}\frac{\partial x}{\partial \theta} + \frac{\partial u}{\partial y}\frac{\partial y}{\partial \theta} = -r\sin\theta\frac{\partial u}{\partial x} + r\cos\theta\frac{\partial u}{\partial y} $$

you should interpret $\partial u/\partial x$ as the derivative of $u$ with respect to its first argument. Similarly for $y$