Question

suppose that f and g are functions such that f(g(x))=x
for every x, and that f'(x)=1/x. prove that g'(x)=x by differentiating f(g(x))

Answer 1

\[f(g(x))=x \rightarrow f'(g(x))g'(x)=1\rightarrow (1/g'(x))g(x)=1\rightarrow g'(x)=g(x)\]
therefore \[g(x)=e^{x}\]

Answer 2

well played sir. and thank you