### Chain Rule & Integration by Substitution

Many ways problem

# Can you find... chain rule edition Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

## Suggestion

Can you find …

1. … functions $f(x)$ and $g(x)$ so that $f(g(x))$ has stationary points when $x=-1$ and $x=5$?

2. … a function $g(x)$ so that $\ln{g(x)}$ has a local minimum?

3. … functions $f(x)$ and $g(x)$ so that $f(g(x))$ has no stationary points?

The derivative of $f(g(x))$ is the product of $f'(g(x))$ and $g'(x)$, provided these are defined.

• When is a product zero?

• When is a product positive or negative?

As well as making sure that $f(g(x))$ is defined, we need to make sure that $f'(g(x))$ and $g'(x)$ are defined, so we need to keep an eye on the domains, ranges and derivatives of $f(x)$ and $g(x).$

Here are a few more suggestions for separate parts of the problem.