Chain rule with g(x) + 1

Question

If y=f(x + g(x))

Then using the chain rule is it correct to state the derivative of y as below?

y'= f'(x + g(x)).(x + g(x))' = (1 + f'(g(x))).(1 + g'(x))

Answer 1

No:\begin{align}y'&=f'\bigl(x+g(x)\bigr).\bigl(x+g(x)\bigr)'\\&=f'\bigl(x+g(x)\bigr).\bigl(1+g'(x)\bigr).\end{align}

Answer 2

If $\,y= f(x+g(x))$

Then $\,\frac{dy}{dx}=f'(x+g(x)).(1+g'(x))$

Answer 3

Your second step is incorrect.

Let $z=x+g(x)$. Then using the Chain Rule, $$\begin{align}y=f(z)&\implies\frac{dy}{dx}=\frac{dy}{dz}\cdot\frac{dz}{dx}\\&\implies\frac{dy}{dx}=f'(z)\cdot(1+g'(x))\\&\implies\boxed{\frac{dy}{dx}=(1+g'(x))f'(x+g(x))}\end{align}$$