Chapter 13.5, Problem 67ES

Derive the identity $\frac{d}{dx}{\displaystyle {\int }_{h\left(x\right)}^{g\left(x\right)}f\left(t\right)dt=f\left(g\left(x\right)\right)g^{\prime }\left(x\right)-f\left(h\left(x\right)\right)h^{\prime }\left(x\right)}$ by letting $u=g\left(x\right)\text{and}\upsilon =h\left(x\right)$ and then differentiating the function $F\left(u,\upsilon \right)={\displaystyle {\int }_{\upsilon }^{u}f\left(t\right)dt}$ with respect to x.