condition to apply integration by parts?

I have two real functions $f, g$ defined on $[a,b]$. Both functions are differentiable on the open interval $(a,b)$.Is this enough to apply integration by parts$$\int_{a}^b f(x) g(x) dx = f(b) G(b) - f(a) G(a) - \int_a^b f'(t) G(t) dt, $$where $G(t) = \int_{a}^t g(x) dx$.My concern is that in the integral expression $f'$ may not be well defined at the end points... Does this matter? Thank you!