TITLE: Differentiation under the integral sign??
QUESTION [5 upvotes]: When I tried to show this, I didn't get the integral of the derivative - only the other terms, and I have no idea why. Here's the working I have;
$\dfrac{d}{dx} \displaystyle\int_{a(x)}^{b(x)} f(x,t)\ dt$
Assuming $f$ integrable with antiderivative $F$:
$\dfrac{d}{dx} \displaystyle\int_{a(x)}^{b(x)} f(x,t)\ dt=\dfrac{d}{dx} [F(x,b(x))-F(x,a(x))]$
By chain rule:
\begin{align}
\dfrac{d}{dx} \displaystyle\int_{a(x)}^{b(x)} f(x,t)\ dt=& \left(\dfrac{\partial F}{\partial x}\dfrac{\partial x}{\partial x} + \dfrac{\partial F}{\partial b}\dfrac{\partial b}{\partial x}\right) - \left(\dfrac{\partial F}{\partial x}\dfrac{\partial x}{\partial x} + \dfrac{\partial F}{\partial a}\dfrac{\partial a}{\partial x}\right)\\
=& \dfrac{\partial F}{\partial b}\dfrac{\partial b}{\partial x}-\dfrac{\partial F}{\partial a}\dfrac{\partial a}{\partial x}\\
=&f(b)b'(x)-f(a)a'(x)
\end{align}
Where did the integral term disappear? I looked at other proofs online, but they're all that insanely ugly infinitesimal argument!

REPLY [5 votes]: You've ignored the points at which you evaluated those expressions. When you write:
$$ \left(\dfrac{\partial F}{\partial x}\dfrac{\partial x}{\partial x} + \dfrac{\partial F}{\partial b}\dfrac{\partial b}{\partial x}\right) - \left(\dfrac{\partial F}{\partial x}\dfrac{\partial x}{\partial x} + \dfrac{\partial F}{\partial a}\dfrac{\partial a}{\partial x}\right)$$
the $\dfrac{\partial F}{\partial x}$ terms don't cancel out, because they're really $\left(\dfrac{\partial F}{\partial x}\right)(x,b(x))$ and $\left(\dfrac{\partial F}{\partial x}\right)(x,a(x))$.