# How do you find the derivative of #y=cos(x^5)# ?

##### 1 Answer

*For this, we will need the Chain rule. The chain rule states that when confronted with a function #f(x) = g(h(x))#, our derivative #f'(x) = (h'(x)) g'(h(x))#. Using this rule, along with the Power Rule and the formula for the derivative of a cosine function, we can find that if #y(x) = cos(x^5), y'(x) = (5x^4)(-sin(x^5))#*,

In the interests of brevity, we will skip a proof of the Chain Rule. However, for those interested, a simple web search for "chain rule proof" will turn up several good results on the first few pages. It is recommended one sticks to the PDF ones, those seem to be more tidily designed.

It is worth noting, however, that even when dealing with a function where

For a more in depth look at your example, let us find the quantities represented by

From the formula

The power rule quickly tells us that

There exists a proof for this (Euler's Formula), but the proof will not be given here. Suffice to say that this shows us that for