I'm trying to comprehend the following result, which is required for fractional calculus:
Let $w(x,y)$ and $f(z)$ be two real functions, such that they both vanish at a point $a$. Then the following relation holds: $$\frac{\text d}{\text dx} \int_a^xw(x,y)f(y)\text dy=w(x)f(x)+\int_a^xf(y)\frac{\partial{w(x,y)}}{\partial{x}}\text dy.$$
My intuition for a proof points towards the integration by parts of a product of functions. But there might be a sign error if that's the case. The reference for this is "Construction & Physical Applications Of The Fractional Calculus", Nicholas Wheeler, 1997.
Thank you for the help!
This is called Leibniz integral rule. In general
$$ {\displaystyle {\frac {d}{dx}}\left(\int _{a(x)}^{b(x)}f(x,t)\,dt\right)=f{\big (}x,b(x){\big )}\cdot {\frac {d}{dx}}b(x)-f{\big (}x,a(x){\big )}\cdot {\frac {d}{dx}}a(x)+\int _{a(x)}^{b(x)}{\frac {\partial }{\partial x}}f(x,t)\,dt.} $$