Question

How do you find the derivative of `y=cos(x^2)` ?

Answer 1

We will need to employ the chain rule.

The chain rule states:

`d/dx[f(g(x))] = d/(d[g(x)])[f(x)] * d/dx[g(x)]`

In other words, just treat `x^2` like a whole variable, differentiate the outside function first, then multiply by the derivative of `x^2`.

We know that the derivative of `cosu` is `-sin u`, where `u` is anything - in this case it is `x^2`. And the derivative of `x^2` is `2x`.

(if those identities look unfamiliar to you, I may direct you to this page or this page, which have videos for the derivative of `cosu` and the power rule, respectively)

Anyhow, by the power rule, we now have:

`d/dx[cos(x^2)] = -sin(x^2) * 2x`

Simplify a bit:

`d/dx[cos(x^2)] = -2xsin(x^2)`