limits of integration and derivative I have an integral that gives
$$\left[\frac{d}{dx}[f(x)]\right]_a^b$$

is it possible in general to claim that this is equal to 
$$\frac{d}{dx}\left(\left[f(x)\right]_a^b\right)\text{ ?}$$

If not in general, are there any sufficient conditions to claim this?

Thanks!!
The first expression is

$$
f'(x)|^b_a = f'(b) - f'(a)
$$
whereas the second one, since neither $a$ nor $b$ are functions of $x$, means
$$
\frac{d}{dx} \left[f(b)-f(a)\right] = 0
$$

So necessary and sufficient condition for equality is $f'(b) = f'(a)$, and one sufficient condition is if $f'$ is constant (so $f$ is linear).
On the left hand side we have: $\left[\dfrac{\mathrm d}{\mathrm d x} f(x)\right]_a^b = \dfrac {\mathrm d f(b)}{\mathrm d b}-\dfrac{\mathrm d f(a)}{\mathrm d a}$.

On the right hand side we have: $\dfrac{\mathrm d}{\mathrm d x}{\big[f(x)\big]}^b_a = \dfrac{\mathrm d}{\mathrm d x}\big(f(b)-f(a)\big) = 0$

In general, the left hand side is not necessarily equal to the right hand side.

One counter example is simply, $f(x)=x^2, a=0, b=1$.  Then $\left[\dfrac{\mathrm d x^2}{\mathrm d x} \right]_0^1 = 2$.