Question

Find $\frac{d}{dx}\left({\cos }^{2}x\right)$ by using chain rule.

Answer 1

Compute the required value:

We know that, according to the chain rule formula, $\frac{dy}{dx}=\frac{dy}{du}\times \frac{du}{dx}$

$$\begin{gathered} \Rightarrow \frac{d}{dx}{\cos }^2\left(x\right)=\frac{d\cos {\left(x\right)}^2}{d\cos \left(x\right)}\times \frac{d\cos \left(x\right)}{dx} \\ =2\cos \left(x\right)\times \left(-\sin \left(x\right)\right) \\ =-\sin \left(2x\right) \end{gathered}$$

Hence, the required solution is $-\sin \left(2x\right)$.