Differentiate an integral of two variables with respect to a function

Question

I have a known function of two variables $f = f(x_0,x)$, and another unknown function of one variable $w=w(x)$. They are related to each other by the equation:

\begin{equation} L(x) = \int_{a}^{b} w(x_0) f(x_0,x) dx_0 \end{equation}

Where $a$ and $b$ are constants. What is the formula for $\frac{\partial L(x)}{\partial w(x)}$?

Answer 1

Essentially you have \begin{align*} \frac{d}{ds}\bigg\vert_{s=0}L(w+sv) &=\int_a^b\frac{d}{ds}\bigg\vert_{s=0}\big(w(x_0)+sv(x_0)\big)f(x_0,x)dx_0\\ &=\int_a^bv(x_0)f(x_0,x)dx_0 \end{align*} as the directional derivative at $w$ in direction $v$. The rest depends on your notation.