§2.5 The chain rule

Introduction

The chain rule

With $f(x) = \sin(x)$ and $g(x) = x^2$, consider $s(x) = 3x^2 - 5\sin(x)$, $p(x) = x^2\sin(x)$, and $q(x) = \frac{\sin(x)}{x^2}$. Which list names the needed rules?

Using the identity $\sin(2x) = 2\sin(x)\cos(x)$, the product rule gives $\frac{d}{dx}[\sin(2x)] = 2(\cos^2(x) - \sin^2(x))$. Simplified, this equals

For $h(x) = \cos(x^4)$, which expression is $h'(x)$?

The function $\tan(2^x)$ is best described as

For $p(x) = \sqrt{\tan(x)}$, which expression is $p'(x)$?

For $s(x) = 2^{\sin(x)}$, which expression is $s'(x)$?

For $m(x) = (\sec(x) + e^x)^9$, which expression is $m'(x)$?

For $r(x) = (\tan(x))^2$, compute $r'(\frac{\pi}{4})$.

For $z(x) = \cot^5(x)$, compute $z'(\frac{\pi}{4})$.

For a composition $C(x) = f(g(x))$ with both pieces differentiable, $C'(x) = f'(g(x)) \cdot g'(x)$.