What is the derivative of this trig function?

Question

What is the derivative of $y=3x-5\cos^2(\pi x)$ ? What rule do I start with?

Answer 1

You'll need the chain-rule for the second term.

If given $y=3x-5\cos^2(\pi x)$, $$y' = 3 - 2\cdot 5\cdot \pi \cos(\pi x)(-\sin(\pi x)) = 3 + 5\pi(2\sin(\pi x)\cos(\pi x)) = 3 + 5\pi\sin(2\pi x)$$

There seems to have been some confusion, given the last edit, prior to my editing to return the question to its original statement:

In case you want to calculate the derivative of the function $$y = 3x - 5\cos\Big((\pi x)^2\Big)$$ then you still need the chainrule:

In this case, the deriviative would be $$\begin{align} y' & = 3 - \left[-5 \sin\Big((\pi x)^2\Big)\right]\cdot \frac{d}{dx}\Big((\pi x)^2 \Big) \\ \\ & = 3 - \left[-5 \sin\Big((\pi x)^2\Big)\right]\cdot \frac{d}{dx}\Big(\pi^2 x^2 \Big)\\ \\ & = 3 + 5 \sin\Big((\pi x)^2\Big)\cdot (2\pi^2 x) \\ \\ & = 3 + 10\pi^2 x\cdot \sin\Big((\pi x)^2\Big)\end{align}$$