Question

What is the derivative of ${\cos }^{2}x$?

Answer 1

Determine the derivative of ${\cos }^{2}x$.

Let $y={\cos }^{2}x$
Differentiating on both sides, with the respect to $x$, we get,
$$\begin{gathered} \frac{\mathrm{d}y}{\mathrm{d}x}=\frac{\mathrm{d}}{\mathrm{d}x}({\cos }^{2}x) \\ \Rightarrow =\frac{\mathrm{d}}{\mathrm{d}x}(\cos x{)}^2 \\ \Rightarrow =2.\cos x\left(\frac{\mathrm{d}}{\mathrm{d}x}\cos x\right) \\ \Rightarrow =2\cos x(-\sin x) \\ \Rightarrow =-2\sin x\cos x \\ \Rightarrow =–\sin 2x \end{gathered}$$
Hence, the required derivative of is $\frac{\mathrm{d}}{dx}\left({\cos }^{2}x\right)=-\sin 2x$.