Chain Rule applied to Trig Functions 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 $2~ \pi^{2} x$
 A: 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 $\sin\Bigl((\pi x)^2\Bigr)$.
A: 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. 
A: The derivative is $\pi \cos(\pi x)$.
