Trouble applying the fundamental theorem of Calculus Let $\psi(t)=\phi(t_0)*\exp(\int_{t_0}^t b(r)dr) + \int_{t_0}^t a(s) \exp(\int_{s}^t b(r)dr)ds$
Now I know already ( because this is from a differential equation example in a book ) that $$\psi'(t)= a(t)+ b(t)*\psi(t)$$ but I can't seem to differentiate correctly. If I differentiate $\psi(t)$  I should get 
$$\psi'(t)=\phi(t_0)*\exp(\int_{t_0}^t b(r)dr)* b(t)+ \frac{d}{dt}\int_{t_0}^t a(s) \exp(\int_{s}^t b(r)dr)ds$$
and then use FTC for the last summand, but "plugging in" $t$ for every $s$ is wrong here because of the $t$ in the last integral and it surely gives me a wrong result if I tried to. I think I have to use some kind of chain rule here, but I need some help with the details! 
How do I differentiate the last summand?
 A: Hint: Use the definition of derivative to find the derivative of a general form:
$$\dfrac d{dt} \int_0^t f(t,s)g(s) \, ds = \lim_{h \to 0} \dfrac 1h \left( \int_0^{t+h} f(t+h,s)g(s) \, ds - \int_0^t f(t,s)g(s) \, ds \right)$$
I think the general rule is, like you said, some kind of product rule. If I recall, it is:
$$\dfrac d{dt} \int_0^t f(t,s)g(s) \, ds = \int_0^t \dfrac{\partial f}{\partial t}(t,s)g(s) \, ds + f(t,t)g(t)$$
A: A slightly more expository way to view this could be to recall that, by properties of integrals, we have:
$$ \begin{align}e^{\int_{s}^{t}b(r)dr} & =e^{(\int_{0}^{t}b(r)dr + \int_{s}^{0}b(r)dr)}\\\\ & =e^{\int_{0}^{t}b(r)dr}e^{ \int_{s}^{0}b(r)dr} \end{align}$$
(I used $0$ here but any constant would work.)
The point of this is then to recognize that $e^{\int_{0}^{t}b(r)dr}$ is independent of $s$, and can be treated as a constant; i.e. pulled out of the integral along $s $:
$$ \begin{align}\int_{t_0}^{t}a(s)e^{\int_{s}^{t}b(r)dr}ds & = \int_{t_0}^{t}a(s)e^{(\int_{0}^{t}b(r)dr + \int_{s}^{0}b(r)dr)}ds\\\\ & =\int_{t_0}^{t}a(s)e^{\int_{0}^{t}b(r)dr}e^{ \int_{s}^{0}b(r)dr}ds \\\\ & = e^{\int_{0}^{t}b(r)dr}\int_{t_0}^{t}a(s)e^{ \int_{s}^{0}b(r)dr}ds\\\\ & = f(t)\int_{t_0}^{t}g(s)ds\end{align}$$
Differentiation is now more manageable. 
Hope this helps.  
