I understand how to do differentiation when the elements are individual functions, but am having trouble applying the concepts with nested functions.
For example, given $(\cos \pi x + \sin \pi y)^5= 24$ $$\frac{d}{dx}\left[\frac{d}{dx}(\cos \pi x)+\frac{d}{dx} (\sin \pi y)\right]^5=\frac{d}{dx}24$$
Following the chain rule of ${f}'(g(x))\cdot {g}'(x), $
$$5[(\cos(\pi x) +\sin (\pi y)]^4 \cdot [-\pi \sin(\pi x) + \pi \cos(\pi y)]=0$$
That's where I get stuck. First, as I see it, I have a few options for the next step:
multiply ${g}'(x)$ by 5
multiply $g(x)$ by 4
What about after that? How do I separate the contents inside the power block to move the $x$ functions to the other side of the equation?