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

1 Answer
Aug 6, 2014

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)