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.