Question

Differentiate the following w.r.t. $x$: ${\cos }^2{x}^3$.

Answer 1

Solution:

The differentiation of the given function is computed by using the chain rule.

Let $y={\cos }^{2}u\mathrm{and}u=x^3$

$$\begin{gathered} \frac{dy}{dx}=\frac{dy}{du}·\frac{du}{dx} \\ =\frac{d}{du}{\cos }^{2}u·\frac{d}{dx}{x}^3 \\ =2\cos u·(-\sin u)·3x^2 \\ =-3x^2(2\sin u·\cos u) \\ =-3x^2\sin 2u\left[\because 2\sin x·\cos x=\sin 2x\right] \\ =-3x^2\sin 2x^3\left[\because u=x^3\right] \end{gathered}$$

Hence, the required answer is $-3x^2\sin 2x^3$.