y = (1+cos^2(7x)^3
i know the answer, but can anyone explain the steps of this to me, it would be much appreciated. some good practice for you wiz's out there. thanks alot.
ok, so you should have typednow the chain rule is used to differentiate composite functions, that is, functions formed by plugging in one function into another. so for example, here, instead of just $\displaystyle x^3$ you have $\displaystyle (1 + \cos^2 7x)^3$, so the function $\displaystyle 1 + \cos^2 7x$ is plugged into $\displaystyle x^3$. moreover, within that function is another composite function. we get $\displaystyle \cos^2 7x$ by plugging in the function $\displaystyle \cos 7x$ into $\displaystyle x^2$, and again, we get $\displaystyle \cos 7x$ by plugging in $\displaystyle 7x$ into $\displaystyle \cos x$
so as you see, we have to do the chain rule a lot of times here.
the rule says, $\displaystyle \frac d{dx} f(g(x)) = f'(g(x)) \cdot g'(x)$
so to differentiate a composite function, you differentiate the outer function as if the inside function was a single variable. then, to compensate for the fact that it really wasn't a single variable, you multiply by the derivative of the inside function.
so here goes:
$\displaystyle \frac d{dx} (1 + \cos^2 7x )^3 = 3(1 + \cos^2 7x)^2 \cdot \underbrace{2 \cos 7x \cdot -7 \sin 7x}_{\text{derivative of } 1 + \cos^2 7x \text{ also by the chain rule}} $ $\displaystyle = -21(1 + \cos^2 7x)^2 \cdot 2 \sin 7x \cos 7x = -21 \sin (14x)(1 + \cos^2 7x)^2$