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Given $f(x)= \sin(\pi x)^{2}$, find the derivative.

Using the chain rule my work is as follows: (\sin(\pi x)^2)' becomes $2 \sin(\pi x) \cdot \frac{d}{dx}(\sin(\pi x)$ The derivative of sin is cos, thus $2 \sin(\pi x) \cdot \cos(\pi x) \cdot \frac{d}{dx}(\pi x)$ The derivative of $\pi x$ is $\pi$, and the equation stretches to

$2 \sin(\pi x) \cos(\pi x) \pi == 2 \pi \sin(\pi x) \cos(\pi x)$

However, the book states the answer as $2 \pi^{2}~ x ~\cos(\pi x)^{2}$ and that definitely doesn't match my result. Where did I go wrong?

EDIT

Thanks to Arturo Madigan, Jonas Meyer, et al for their help.

I re-did the problem based on having $(\pi x)^{2}$, having the exponent rather than the sin function, and it seems I have a missing exponent as well.

Differentiating the terms of the function via the chain rule, I get $(\pi x)^{2} [\frac{d}{dx}sin] \cdot \frac{d}{dx}(\pi x)^{2}$

$ cos(\pi x)^{2} \cdot 2\pi x= 2~\pi~ x cos~(\pi x)^{2}$

According to the book answer, $2~\pi x$ should actually be $2~ \pi^{2} x$

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    @Jonas: I plead guilty.2011-03-07

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Given the answer, the question was for the derivative of $\sin\Bigl( (\pi x)^2\Bigr)$; instead, you computed the derivative of $\Bigl(\sin(\pi x)\Bigr)^2$.

If you were computing the derivative of the latter, then your computations are correct; the derivative is $2\pi\sin(\pi x)\cos(\pi x)$.

But if you were asked for the derivative of $\sin\Bigl((\pi x)^2\Bigr)$, then of course you were looking at the wrong function, and that's why the answers don't match.

It's possible you had "$\sin(\pi x)^2$" and interpreted this as $(\sin(\pi x))^2$; usually, $\sin^2(\pi x)$ is used for the latter, so "$\sin(\pi x)^2$" would be interpreted as $\sin\Bigl((\pi x)^2\Bigr)$.

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    I'm reminded of a sign that was supposedly posted in a newsroom (back in the day of lead type)-"the composing room has an infinite supply of periods to terminate short, complete sentences". Maybe that applies to parentheses, too. We got lots of 'em, so use 'em liberally.2011-03-07
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Think of the function $\sin(\pi x)$ like this: $x \to \pi x \to \sin(\pi x)$. In which case it is intuitively clear that the rate of change of the function should be the multiplication of the rate of changes of $x \to \pi x$ and $y \to \sin (y)$, the latter being evaluated at $y=x$. Hence PEV's answer.

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The derivative is $\pi \cos(\pi x)$.

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    Based on the OP's steps and the answer in the back of the book, I'm pretty sure $\sin(\pi x)$ was not intended. (I elaborate in my comment on the question.)2011-03-07