Notation for function compositions/derivatives

Question

When given $(f \circ g)'(0)$, does it mean to compose the 2 functions first, then take the derivative of the composed functions and evaluate it at $0$, or take the derivative of $g$ first and evaluate it at $0$, then take the derivative of $f$ and evaluate it at the value that the evaluation of $g'(0)$ gave us?

Answer 1

When given $(f\circ g)'(0)$, we can compose the functions first, then take the derivative of the composed function, and evaluate it at $x=0$, ...

Or... recall that this situation occurs whenever we need the chain rule. That is we can use the chain rule to calculate $$(f\circ g)'(x) = (f(g(x)))' = f'(g(x))\cdot g'(x)$$

then simply evaluate the derivative at $x=0$.

Answer 2

The parentheses indicate that the composition happens first, and then the derivative of the composed function is to be taken at the point $x=0$.

Answer 3

$(f \circ g)'(0)$ wants you to get the composite function $(f \circ g)(x) = h(x)$, then find the derivative of $h'(x)$ in $0$. $$(f \circ g)'(0) = D[f(g(0))] = f'(g(0))\cdot g'(0)$$