Corollary 2.10.6
\begin{align*}
\diff{}{x} a^x &= \log a \cdot a^x & \text{for any $a \gt 0$}\\
\diff{}{x} \log_a x &= \frac{1}{x \cdot \log a} & \text{for any $a \gt 0, a \neq 1$}
\end{align*}
where \(\log x\) is the natural logarithm.
We can also now finally get around to computing the derivative of \(a^x\) (which we started to do back in Section 2.7).
Notice that we could have also done the following:
We then process the left-hand side using the chain rule
\begin{align*} f'(x) \cdot \frac{1}{f(x)} &= \log a\\ f'(x) &= f(x) \cdot \log a = a^x \cdot \log a \end{align*}We will see \(\diff{}{x} \log f(x)\) more below in the subsection on “logarithmic differentiation”.
To summarise the results above:
where \(\log x\) is the natural logarithm.
Recall that we need the caveat \(a \neq 1\) because the logarithm base 1 is not well defined. This is because \(1^x = 1\) for any \(x\text{.}\) We do not need a similar caveat for the derivative of the exponential because we know (recall Example 2.7.1)