Answer
False
Work Step by Step
Counter-example: let $ f(x)=6x+4; g(x)=6x+7$. Using the power rule, $f'(x)=6 ; g'(x) =6$. While the derivatives are the same, the functions are clearly not the same.
Explanation: When differentiating, the constant gets lost so two functions that have the same derivative will have the same terms down to and including $x$, but the constant could be different. In fact, two functions that have the same derivative belong to a family of functions but are not necessarily the same function.
