Differentiate the functions with respect to x. ^ 3 . ^ 2 (x^ 5 ) - Vidyadip

Question

Differentiate the functions with respect to x.
$\cos\text{x}^{3}. \sin^{2}(\text{x}^{5})$

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Answer

$\text{Let y} =\cos\text{x}^{3}. \sin^{2}(\text{x}^{5})$
$\therefore \frac{\text{dy}}{\text{dx}} =\cos\text{x}^{3} \frac{\text{d}}{\text{dx}}\sin^{2}(\text{x}^{5})+\sin^{2}(\text{x}^{5})\frac{\text{d}}{\text{dx}}\cos\text{x}^{3}$
$=\cos\text{x}^{3}.2\sin(\text{x}^{5}) \frac{\text{d}}{\text{dx}}\sin(\text{x}^{5})+\sin^{2}(\text{x}^{5})(-\sin\text{x}^{3})\frac{\text{d}}{\text{dx}}\text{x}^{3}$
$=\cos\text{x}^{3}.2\sin\text{x}^{5} .\cos\text{x}^{5}\frac{\text{d}}{\text{dx}}\text{x}^5+\sin^{2}(\text{x}^{5})(-\sin\text{x}^{3})3\text{x}^{2}$
$=\cos\text{x}^{3}.2\sin(\text{x}^{5}) \cos(\text{x}^{5})(5\text{x}^{4})-\sin^{2}(\text{x}^{5})\sin\text{x}^{3}.3\text{x}^{2}$
$=10\text{x}^{4}\cos\text{x}^{3} \sin(\text{x}^{5})\cos(\text{x}^{5})-3\text{x}^{2}\sin^{2}(\text{x}^{5})\sin\text{x}^{3}$

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