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**A Beautiful Mind** $\displaystyle sin^2(x^2) = sin(x^2)cos(x^2)$

I need to know how to get this part.

Here's the problem:

$\displaystyle f(x) = sin^2(x)sin(x^2)sin^2(x^2)$

Now it's obvious that you can use the product rule to differentiate the three of these, which I've done.

It'll look something like this:

$\displaystyle f'(x)g(x)h(x)+f(x)g'(x)h(x)+f(x)g(x)h'(x)$

However, on the last part, I'm only able to get 2 parts of the answer. The $\displaystyle f(x) $and $\displaystyle g(x) $are kept constant while I'm having trouble differentiating the $\displaystyle h'(x)$:

$\displaystyle sin^2(x^2)$

When seemed obvious to me was that you'd bring the power of 2 to the outside and then differentiate, which would lead to $\displaystyle 2sin(x^2)cos(x^2)2x, $but this is not the answer in the back of the book.

$\displaystyle f(x)g(x)h'(x)$ is supposed to be = $\displaystyle sin(x^2)cos(x^2)sin^2xsinx^2$

I had four tutors working on this problem who gave up on it at my college, which I thought was either sad or Michael Spivak (author of this legendary Calculus text) made an error.