How can i derive the following implication out of this theorem:
Theorem (proven): is $f:[a,b] \rightarrow \mathbb{R} $ riemann integrable and $f$ in the point $x_o \in [a,b]$ Continious, then is $F(x)= \int_a^x f(t)dt $ in that point diffrentiable and $F'(x_o)=f(x_o)$
Implication: for $f:[a,b]\times[c,d] \rightarrow \mathbb{R}$ Continious and all the partial derivates exist and are continious, we define
$F(y) = \int_a^b f(x,y)dx$
We want to show that $F'(y)= \int_a^b \frac{\partial f}{\partial y}(x,y) dx$