Compositions and choosing derivative rules

Understand the idea

Identify outer and inner functions before differentiating. Differentiate the outer function at the inner expression, then multiply by the inner derivative. Several layers require several factors. Combine the chain rule with product or quotient rules when the top-level structure requires them. Exponential bases other than $\displaystyle e$ contribute a logarithmic factor. For a logarithm with base $\displaystyle a>0$ , $\displaystyle a\ne1$ , differentiate $\displaystyle \log_a x$ as $\displaystyle 1/(x\ln a)$ on $\displaystyle x>0$ .

(f\circ g)'(x)=f'(g(x))g'(x),\quad (a^x)'=a^x\ln a,\quad(\ln|u|)'=\frac{u'}u

Worked example

Differentiate $\displaystyle y=\sin(x^2+1)$ .

The outer function is sine and its derivative is cosine. The inner derivative is

\displaystyle 2x

, so

\displaystyle y'=2x\cos(x^2+1)

. Keep the original inner expression inside cosine.

Try it yourself

Find $\displaystyle d(\ln(1+x^2))/dx$ .

Show solution and reasoning

The chain rule gives $\displaystyle 2x/(1+x^2)$ . The logarithm is defined for every real $\displaystyle x$ here.