We have found that the gradient function of \(a^x\) can be written as \[f'(x) = a^x\times f'(0),\]

i.e. the gradient function is the function itself multiplied by a constant, and this constant is the gradient of \(a^x\) at \(x = 0\). There will be a special case when \(f'(0) = 1\) as then the gradient function of \(a^x\) would be itself, \(a^x\).Before using the applet, look at the values given. Approximately what value do you think \(a\) will take to give \(f'(0) \approx 1?\)

Use the slider to find the value of \(a\) when \(f'(0) \approx 1\).

Does the gradient function look as you expected?