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Question Number 122434 by benjo_mathlover last updated on 17/Nov/20

$$Iff\left(x\right)={\int }_a^{\left(\underset{a}{\stackrel{x^3}{\int }}\frac{dt}{1+\sin {}^{2}t}\right)}\left(\frac{dt}{1+\sin {}^{2}t}\right)$$$$thenf{}^{\prime }\left(x\right)?$$

Answered by liberty last updated on 17/Nov/20

$$\mathrm{f}{}^{\prime }\left(\mathrm{x}\right)=\frac{{\left({\mathrm{x}}^3\right)}^{\prime }}{(1+\sin {}^2\left({\mathrm{x}}^3\right)).(1+\sin {}^2\left(\underset{1}{\stackrel{{\mathrm{x}}^3}{\int }}\frac{\mathrm{dt}}{1+\sin {}^2\mathrm{t}}\right))}$$$$\mathrm{f}{}^{\prime }\left(\mathrm{x}\right)=\frac{{3\mathrm{x}}^2}{(1+2\sin {}^2\left({\mathrm{x}}^3\right)).(1+\sin {}^2\left(\underset{1}{\stackrel{{\mathrm{x}}^3}{\int }}\frac{\mathrm{dt}}{1+\sin {}^2\mathrm{t}}\right))}$$

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