derivatives - Why is this the first step of differentiating [a^u(x)]

Following this enter link description here, where we're performing the derivative
$$\frac{d}{dx}5^{x\cos(x)},$$

The first step of the differentiation is:

$[a^u(x)]'$ = $\ln(a) \,a^u(x)\, u'(x).$

I’m quite confused by the intuition. Where did the $\ln$ come from? Why do we need to multiply by $a^u(x)?$

Would appreciate a step by step breakdown of why this is the first step. Is this just another differentiation rule to remember (like chain rule, product rule) that I should instinctively know when performing differentiation, or is there some manipulation that lead to the equation above? This is a new topic for me, apologies.

Answer

We have that

$$a^{u(x)}=e^{u(x)\cdot \log a}$$

and then

$$(a^{u(x)})'=(e^{u(x)\cdot \log a})'=e^{u(x)\cdot \log a}\cdot (u(x) \cdot \log a)'=a^{u(x)}\cdot u'(x) \cdot \log a$$