Leibniz' Rule For Differentiating Integrals

If the endpoint of an integral is a function of x rather than simply x, then we need to use the Chain Rule together with part 1 of the Fundamental Theorem of Calculus to calculate the derivative of the integral. According to the Chain Rule,

if \frac{\mathrm{d}}{\mathrm{dx}} \mathrm{A}(x)=\mathrm{f}(x), \text { then } \frac{\mathrm{d}}{\mathrm{dx}} \mathrm{A}\left(x^{2}\right)=\mathrm{f}\left(x^{2}\right) \cdot 2 x and, applying the Chain Rule to the derivative of the integral,

\frac{\mathrm{d}}{\mathrm{dx}}\left(\int_{\mathrm{a}}^{\mathrm{g}(x)} \mathrm{f}(t) \mathrm{dt}\right)=\frac{\mathrm{d}}{\mathrm{dx}} \mathrm{A}(\mathrm{g}(x))=\mathrm{f}(\mathrm{g}(x)) \cdot \mathrm{g}^{\prime}(x)


If f is a continuous function and \mathrm{A}(x)=\int_{\mathrm{a}}^{x} \mathrm{f}(t) \mathrm{dt}.

then \frac{\mathrm{d}}{\mathrm{dx}}\left(\int_{\mathrm{a}}^{x} \mathrm{f}(t) \mathrm{dt}\right)=\frac{\mathrm{d}}{\mathrm{dx}} \mathrm{A}(x)=\mathrm{f}(x) (Fundamental Theorem, Part I)

and, if g is differentiable, \frac{\mathrm{d}}{\mathrm{d} \mathrm{x}}\left(\int_{\mathrm{a}}^{\mathrm{g}(x)} \mathrm{f}(t) \mathrm{dt}\right)=\frac{\mathrm{d}}{\mathrm{dx}} \mathrm{A}(\mathrm{g}(x))=\mathrm{f}(\mathrm{g}(x)) \cdot \mathrm{g}^{\prime}(x) (Leibniz' Rule)


Example 6: Calculate \frac{\mathrm{d}}{\mathrm{dx}}\left(\int_{\mathrm{a}}^{5 x} t^{2} \mathrm{dt}\right), \frac{\mathrm{d}}{\mathrm{dx}}\left(\int_{\mathrm{a}}^{x^{2}} \cos (u) \mathrm{du}\right), \frac{\mathrm{d}}{\mathrm{dw}}\left(\int_{\mathrm{a}}^{\sin (w)}{\mathrm{z}}^{3} \mathrm{dz}\right).

Solution: \begin{gathered} \frac{\mathrm{d}}{\mathrm{dx}}\left(\int_{\mathrm{a}}^{5 x} \mathrm{dt}\right)=(5 x)^{2} \cdot 5=125 x^{2} \cdot \frac{\mathrm{d}}{\mathrm{d} \mathrm{x}}\left(\int_{\mathrm{a}}^{x^{2}} \cos (u) \mathrm{du}\right)=\cos \left(x^{2}\right) \cdot 2 x=2 x \cdot \cos \left(x^{2}\right) \\ \end{gathered}

\begin{aligned} &\frac{\mathrm{d}}{\mathrm{dw}}\left(\int_{\mathrm{a}}^{\sin (w)} z^{3} \mathrm{dz}\right)=(\sin (w))^{3} \cos (w)=\sin ^{3}(w) \cos (w) \\ & \end{aligned}


Practice 5: Find \frac{\mathrm{d}}{\mathrm{dx}}\left(\int_{0}^{x^{3}} \sin (t) \mathrm{dt}\right).