# JEE Main & Advanced Mathematics Differentiation Differentiation of Integral Function

## Differentiation of Integral Function

Category : JEE Main & Advanced

If ${{g}_{1}}(x)$ and ${{g}_{2}}(x)$ both functions are defined on $[a,\,\,b]$ and differentiable at a point $x\in (a,b)$ and $f(t)$ is continuous for ${{g}_{1}}(a)\le f(t)\le {{g}_{2}}(b)$, then

$\frac{d}{dx}\int_{{{g}_{1}}(x)}^{{{g}_{2}}(x)}{f(t)dt}=f[{{g}_{2}}(x)]{{{g}'}_{2}}(x)-f[{{g}_{1}}(x)]{{{g}'}_{1}}(x)$

$=f[{{g}_{2}}(x)]\frac{d}{dx}{{g}_{2}}(x)-f[{{g}_{1}}(x)]\frac{d}{dx}{{g}_{1}}(x)$.

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