Question
Differentiate the function with respect to x : $\cos {x^3}{\sin ^2}\left( {{x^5}} \right)$
$\therefore \frac{{dy}}{{dx}} = \cos {x^3}\frac{d}{{dx}}{\sin ^2}\left( {{x^5}} \right) + {\sin ^2}\left( {{x^5}} \right)\frac{d}{{dx}}\cos {x^3}$
$= \cos {x^3}.2\sin \left( {{x^5}} \right)\frac{d}{{dx}}\sin \left( {{x^5}} \right) + {\sin ^2}\left( {{x^5}} \right)\left( { - \sin {x^3}} \right)\frac{d}{{dx}}{x^3}$
$= \cos {x^3}.2\sin \left( {{x^5}} \right)\frac{d}{{dx}}\sin \left( {{x^5}} \right) + {\sin ^2}\left( {{x^5}} \right)\left( { - \sin {x^3}} \right)3{x^2}$
$= \cos {x^3}.2\sin \left( {{x^5}} \right)\cos \left( {{x^5}} \right)\left( {5{x^4}} \right) - {\sin ^2}\left( {{x^5}} \right)\sin {x^3}.3{x^2}$
$= 10{x^4}\cos {x^3}\sin \left( {{x^5}} \right)\cos \left( {{x^5}} \right) - 3{x^2}{\sin ^2}\left( {{x^5}} \right)\sin {x^3}$
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