Is this expression of the chain rule correct

Question

Supposed general form, where $f$ and $g$ are functions: $$\frac{d}{dx}f(g(x))$$

$$=\frac{d}{dx}f(x)\times\frac{d}{df(x)}g(f(x))$$

Specific case:

$$\frac{d}{dx}ln(e^x)=\frac{d}{dx}e^x\times\frac{d}{d(e^x)}\ln(e^x)$$

Please prioritize on the general form

Answer 1

What you have in the example is correct.

$\frac{d}{dx}ln(e^x)=$$\frac{d}{dx}e^{x}\times\frac{d}{d(e^x)}\ln(e^x)\\e^x \times \frac 1{e^x}$

What you have in the general should read:

$\frac{d}{dx}f(g(x)) = \frac{d}{dx} g(x) \times \frac{d}{dg(x)}f(g(x))$

The notation is ususally written $\frac{d}{dx}f(g(x)) = \frac{df}{dg}\frac{dg}{dx}$