Question

If $\varphi \left(x\right)=\stackrel{\sqrt{x}}{\underset{1/x}{\int }}\sin \left(t^2\right)dt$, then find the value of ${\varphi }^{\prime }\left(1\right)$.

Answer 1

$\varphi \left(x\right)={\int }_{\frac{1}{x}}^{\sqrt{x}}\sin \left(t^2\right)dt$

using leibitz rule we, get

${\varphi }^{\prime }\left(x\right)=sin\left(\right(\sqrt{x}{)}^2)\frac{d}{dx}\sqrt{x}-\sin \left({\left(\frac{1}{x}\right)}^2\right)\frac{d}{dx}\frac{1}{x}$

${\varphi }^{\prime }\left(x\right)=\sin x.\frac{1}{2}{x}^{-1/2}-\sin \left(\frac{1}{x^2}\right)\left(\frac{-1}{x^2}\right)$

${\varphi }^{\prime }\left(1\right)=\frac{\sin 1}{2}+\frac{\sin 1}{1}$

${\varphi }^{\prime }\left(1\right)=\frac{3}{2}\sin 1$