# Teacher Notes

### 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.