Leibnitz's Rule
Category : JEE Main & Advanced
(1) If \[f(x)\] is continuous and \[u(x),\,\,v(x)\] are differentiable functions in the interval \[[a,\,\,b]\] then,
\[\frac{d}{dx}\int_{u(x)}^{v(x)}{f(t)dt=f\{v(x)\}\frac{d}{dx}}\{v(x)\}-f\{u(x)\}\frac{d}{dx}\{u(x)\}\].
(2) If the function \[\varphi \,(x)\] and \[\psi \,(x)\] are defined on \[[a,\,\,b]\] and differentiable at a point \[x\in \,(a,b),\] and \[f(x,t)\] is continuous, then, \[\frac{d}{dx}\,\left[ \int_{\varphi (x)}^{\psi (x)}{{}}f(x,t)\,dt \right]\]\[=\int_{\varphi (x)}^{\psi (x)}{\frac{d}{dx}}\,f(x,t)\,dt+\left\{ \frac{d\,\psi (x)}{dx} \right\}\,f(x,\psi (x))\]\[-\left\{ \frac{d\varphi (x)}{dx} \right\}f(x,\,\varphi (x))\].
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