\begin{array}{rcl}(F(x+h)-F(x))/h&=&
\displaystyle
\frac 1h\left(\int_{a(x+h)}^{b(x+h)}f(x+h,t)\,dt-\int_{a(x)}^{b(x)}f(x,t)\,dt\right)
\vcspace{0.5em}\\
&=&\displaystyle
\frac 1h\left(\int_{a(x)}^{b(x)}f(x+h,t)\,dt+\int_{b(x)}^{b(x+h)}f(x+h,t)\,dt-\int_{a(x)}^{a(x+h)}f(x+h,t)\,dt-\int_{a(x)}^{b(x)}f(x,t)\,dt\right)
\vcspace{0.5em}\\
&=&\displaystyle
\int_{a(x)}^{b(x)}\frac{f(x+h,t)-f(x,t)}{h}\,dt+\frac 1h\int_{b(x)}^{b(x+h)}f(x+h,t)\,dt-\frac 1h\int_{a(x)}^{a(x+h)}f(x+h,t)\,dt