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

1 Answer
Jacob F.
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)#

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