Question

The derivative of $\cos \left(x^3\right) \sin ^2\left(x^5\right)$ is

(A) $10 x^4 \sin \left(x^5\right) \cos \left(x^5\right) \cos \left(x^3\right)$$-3 x^2 \sin ^2\left(x^5\right) \sin \left(x^3\right)$

(B) $10 x^4 \sin \left(x^5\right) \cos \left(x^5\right)-3 x^2 \sin \left(x^3\right) \sin ^2\left(x^5\right)$

(C) $10 \sin \left(x^5\right) \cos \left(x^5\right) \cos \left(x^3\right)-3 x^2 \sin ^2\left(x^5\right) \sin \left(x^3\right)$

(D) None of the above

Answer 1

Let $y=\cos \left(x^3\right) \sin ^2\left(x^5\right)$

$\begin{aligned}\Rightarrow \quad \frac{d y}{d x} & =\cos \left(x^3\right)\left(2 \sin \left(x^5\right) \cos \left(x^5\right) \times 5 x^4\right) \\& +\sin ^2\left(x^5\right)\left(-\sin x^3\left(3 x^2\right)\right) \\= & 10 x^4 \sin \left(x^5\right) \cos \left(x^5\right) \cos \left(x^3\right) \\& -3 x^2 \sin ^2\left(x^5\right) \sin \left(x^3\right)\end{aligned}$

So, The correct option will be (A).