How do you differentiate $sin(x^2)(cos(x^2))$ ?

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Answer 1 · 2015-09-07T13:23:56

$d/dx sin(x^2) cos(x^2) =2xcos(2x^2)$

This problem can also be solved by directly applying the differentiation rules:

$d/dx sin(x^2) cos(x^2)$

First, use the product rule:
$=sin(x^2) d/dx cos(x^2) + cos(x^2)d/dx sin(x^2)$

Then, each derivative can be solved using the chain rule:
$=-sin(x^2) sin(x^2) d/dx x^2 + cos(x^2)cos(x^2)d/dx x^2$

$=-2xsin^2(x^2) + 2xcos^2(x^2)$

At this point, we can simplify the expression by factoring out $2x$ and applying the double angle identity $cos 2theta = cos^2 theta - sin^2 theta$ :

$=2x(cos^2(x^2)-sin^2(x^2))$

$=2xcos(2x^2)$

How do you differentiate sin(x^2)(cos(x^2))sin(x^2)(cos(x^2))?