The discussion centers on applying Leibniz's Rule to differentiate integrals of the form \(\frac{\mathrm{d}}{\mathrm{d} x}\int_{g(x)}^{h(x)} f(x,t) dt\). The established formula derived from Leibniz's Rule states that \(\frac{d}{dx} \int_{g(x)}^{h(x)} f(x,t)dt = \frac{dh}{dx}f(x, h(x)) - \frac{dg}{dx}f(x,g(x)) + \int_{g(x)}^{h(x)} \frac{\partial f(x,t)}{\partial x} dt\). This formula is crucial for understanding how to differentiate integrals with variable limits and integrands that depend on the differentiation variable.
PREREQUISITESStudents of calculus, mathematicians, and educators seeking to deepen their understanding of differentiation techniques involving integrals, particularly those interested in advanced calculus concepts.
mathmadx said:Dear all, a question which has puzzled me for some days:
(Assume that all are differentiable enough times):
Calculate:
[tex] \frac{\mathrm{d} }{\mathrm{d} x}\int_{g(x)}^{h(x)} f(x,t) dt[/tex]