How to prove the following? $\frac{d}{dx}a^x=(\ln a)a^x$

Question

How to prove that the following holds? $$\frac{d}{dx}a^x=(\ln a)a^x.$$ Just a hint will do it.

Answer 1

Hint: $$\dfrac{\mathrm d}{\mathrm dx}a^x=\dfrac{\mathrm d}{\mathrm dx}e^{\ln a\cdot x}.\tag{$a>0$}$$ Now use the chain rule knowing that: $$\dfrac{\mathrm d}{\mathrm dx}e^x=e^x.$$

Answer 2

For $a>0$, we know that $a^x=e^{x\log a}$. Hence to calculate the derivative of $a^x$, you have to calculate the derivative of $e^{g(x)}$ where $g(x)=x\log a$ for $a>0$ which is $g'(x)e^{g(x)}$ using the chain rule.

For $a\le0$, I do not think such a function exists.