The most general form of differentiation under the integral sign states that: if
$f(x, t)$ is a continuous and continuously differentiable (i.e., partial
derivatives exist and are themselves continuous) function and the limits of
integration $a(x)$ and $b(x)$ are continuous and continuously differentiable
functions of $x$, then $$\frac{\mathrm{d}}{\mathrm{d} x} \int_{a(x)}^{b(x)} f(x,
t) \mathrm{d} t=f(x, b(x)) \cdot b^{\prime}(x)-f(x, a(x)) \cdot
a^{\prime}(x)+\int_{a(x)}^{b(x)} \frac{\partial}{\partial x} f(x, t) \mathrm{d}
t$$ Problems based on Leibniz rule are often asked in NET/JEE/JAM/NBHM
examinations. Here we give a few of them. (If you find any mistakes with the
answers, kindly let us know!)
1. [NBHM]Evaluate $f^{\prime}(3)$, where $$ f(x)=\int_{-x}^{x} \frac{1-\mathrm{e}^{-x y}}{y} \mathrm{~d} y \quad, \mathrm{x}>0 $$ Ans: $\frac{2}{3}\left(e^{9}-e^{-9}\right).$
2. [Berkeley Problem]Define $$ F(x)=\int_{\sin x}^{\cos x} e^{\left(t^{2}+x t\right)} d t $$ Compute $F^{\prime}(0)$. Ans: $F^{\prime}(0)=\frac{1}{2}(e-3)$
3. [JAM]Let a be a non zero real number. Then $\lim _{x \rightarrow a} \frac{1}{x^{2}-a^{2}} \int _a^x \sin \left(t^{2}\right)$ dt equals what?
Ans: $\frac{\sin \left(a^{2}\right)}{2 a}$.
1. [NBHM]Evaluate $f^{\prime}(3)$, where $$ f(x)=\int_{-x}^{x} \frac{1-\mathrm{e}^{-x y}}{y} \mathrm{~d} y \quad, \mathrm{x}>0 $$ Ans: $\frac{2}{3}\left(e^{9}-e^{-9}\right).$
2. [Berkeley Problem]Define $$ F(x)=\int_{\sin x}^{\cos x} e^{\left(t^{2}+x t\right)} d t $$ Compute $F^{\prime}(0)$. Ans: $F^{\prime}(0)=\frac{1}{2}(e-3)$
3. [JAM]Let a be a non zero real number. Then $\lim _{x \rightarrow a} \frac{1}{x^{2}-a^{2}} \int _a^x \sin \left(t^{2}\right)$ dt equals what?
Ans: $\frac{\sin \left(a^{2}\right)}{2 a}$.
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