Building blocks

# Chain mapping Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

## Problem

In Mapping a derivative, we saw that we can think of a derivative as a local scaling. What happens if we now want to find the derivative of a composite of two functions?

• What does the mapping diagram of the composite $h(x)=g(f(x))$ look like if $f(x)=x^2+3$ and $g(x)=\sqrt{x}$ (so that $h(x)=\sqrt{x^2+3}$), and we centre our mapping diagram at $1$ on the input numberline?

• How does the diagram change as we zoom in?

• What does this tell us about the derivative $h'(1)$?

• Generalising this, given any composite function $h(x)=g(f(x))$, how can we find $h'(a)$ for a given $a$?

You can download blank composite mapping diagrams with one on a page or two on a page, or you can use the interactivity to explore this.