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Leibnitz's Rule

:
{d \over {dx}}\left[ {\int_{g(x)}^{h(x)} {f(x,t)dt} } \right] = \int_{g(x)}^{h(x)} {{\partial \over {\partial x}}f(x,t)dt + h'(x)f\{ x,h(x)\} - g'(x)f\{ x,g(x)\} }

A simplified situation arises when f depends only on t, making the partial derivative term zero.

{d \over {dx}}\left[ {\int_{g(x)}^{h(x)} {f(t)dt} } \right] = \int_{g(x)}^{h(x)} {{\partial \over {\partial x}}f(t)dt + h'(x)f\{ x,h(x)\} - g'(x)f\{ x,g(x)\} }

\Rightarrow {d \over {dx}}\left[ {\int_{g(x)}^{h(x)} {f(t)dt} } \right] = 0 + h'(x)f\{ x,h(x)\} - g'(x)f\{ x,g(x)\}

\Rightarrow {d \over {dx}}\left[ {\int_{g(x)}^{h(x)} {f(t)dt} } \right] = h'(x)f\{ x,h(x)\} - g'(x)f\{ x,g(x)\}

» Physics, Chemistry & Maths Glossary