Question

Differentiate the function with respect to $x$
$\cos x^3.{\sin }^2\left(x^5\right)$

Answer 1

Let $f\left(x\right)=\cos x^3.{\sin }^2\left(x^5\right)$
$\therefore f^{\prime }\left(x\right)=\frac{d}{dx}[\cos x^3.{\sin }^2(x^5\left)\right]={\sin }^2\left(x^5\right)\times \frac{d}{dx}\left(\cos x^3\right)+\cos x^3\times \frac{d}{dx}\left[{\sin }^2\right(x^5\left)\right]$
$={\sin }^2\left(x^5\right)\times (-\sin x^3)\times \frac{d}{dx}\left(x^3\right)+\cos x^3\times 2\sin \left(x^5\right).\frac{d}{dx}\left[\sin x^5\right]$
$=-\sin x^3{\sin }^2\left(x^5\right)\times 3x^2+2\sin x^5\cos x^3.\cos x^5\times \frac{d}{dx}\left(x^5\right)$
$=-3x^2\sin x^3.{\sin }^2\left(x^5\right)+2\sin x^5\cos x^5\cos x^3\times 5x^4$
$=10x^4\sin x^5\cos x^5\cos x^3-3x^2\sin x^3{\sin }^2\left(x^5\right)$