### Why use this resource?

The aim of the resource is to derive a general form for differentiating \(a^x\) for \(a\) a positive number. The problem is broken down into steps and students are provided with prompting questions to help them.

### Preparation

Students will need to know the laws of logarithms and the derivative of \(e^x\).

### Possible approaches

This could either be used with students who are familiar with the chain rule to build on what they know about \(e^x\), or it could be used as part of an introduction to the chain rule, giving students a chance explore how they might deal with the composite function \(e^{x\ln x}\).

### Key questions

- Which is steeper, \(e^x\) or \(2^x\)?
- How can we rewrite \(y=2^x\) in terms of \(y = e^{kx}\)??
- Why does this help us differentiate \(y=2^x\)?

### Possible extension

Students could make links between the transformations of the graphs \(y=e^{x\ln2}\) and \(y=e^x\), and think about how this would affect the tangent to the curve.