Can anyone explain how to differentiate the Lambert W function?

Question

I'm interested in the differentiation of the Lambert W function $y = xe^x$.

I am unable to understand how to proceed for it.

Answer 1

Hint: Use the product rule for differentiation.

Answer 2

The Lambert W-function is not $y=x e^x$, it is the inverse function of $xe^x$. Since: $$e^{W(x)}W(x) = x \tag{1}$$ by differentiating both sides of $(1)$ with respect to $x$ we get: $$ W'(x)(1+W(x))e^{W(x)} = 1 \tag{2}$$ or: $$ W'(x)(1+W(x)) = \frac{W(x)}{x}\tag{3}$$ hence: $$ W'(x) = \frac{W(x)}{x(1+W(x))}.\tag{4}$$