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.