Worksheet for Section 3.4

Secton 3.4 is about the Chain Rule for derivativess, which allows you to find derivatives of composite functions, of the form $h(x)=f(g(x))$ (so here $h$ is the composition of $f$ and $g$). The Chain Rule says: if $y=f(u)$ is a differentiable function of $u$ and $u=g(x)$ is a differentiable function of $x$ (it is important to keep the variables $u$ and $x$ separate here), then:

$$\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$$

or equivalently:

$$\frac{d}{dx}[f(g(x))]=f'(g(x))\cdot g'(x)$$

(Watch for how the outer function $f(u)$ and the inner function $g(x)$ are used here.) Here are some examples of functions that can be written as composite functions, $y=f(g(x))$ -- in each case, find $y'$:

$$y=\frac1{x^2+1}\qquad y=e^{3x}\qquad y= \sqrt{x^2+3x+5}\qquad y=e^{-\sqrt{x}}$$

The Chain Rule can be used to write much more general versions of some other basic rules -- for example, here is the General Power Rule:

$$\frac{d}{dx}[u^n]=nu^{n-1}\cdot u'$$

(The $u'$ factor is what comes from the Chain Rule -- here $u$ is a function of $x$.) Use this rule to compute the derivative of each of the following functions:

$$f(x)=(4x^2-3x)^4\qquad g(x)=\sqrt[3]{x^2+1}\qquad h(x)=\frac{3}{(2x+1)^2}$$

The Chain Rule can be used repeatedly in cases involving more complicated compositions of functions (like $f(g(h(x)))$). Here is an example:

$$f(x)=\sqrt{1+e^{\sqrt{1+x^2}}}$$

The Chain Rule (along with the Product Rule) also provides an alternative to the Quotient Rule. For the following example compute the derivative both using the Quotient Rule and using the Chain Rule and Product Rule:

$$g(x)=\frac{x^2}{x^2+1}$$