Question

What is the Derivative of $y={\sec }^2\left(x\right)$?

Answer 1

Find the Derivative of $y={\sec }^2\left(x\right)$.

Differentiate both sides w.r.t $x$using chain rule.

$$\begin{gathered} \frac{dy}{dx}=\frac{d}{dx}{\sec }^2\left(x\right) \\ =2\sec \left(x\right)\frac{d}{dx}\sec \left(x\right)\left[\begin{array}{c}\mathrm{chain}\mathrm{rule}:\frac{d}{dx}\left[f\left(g\left(x\right)\right)\right]=\frac{d}{dx}\left[f\left(g\left(x\right)\right)\right]\times \frac{d}{dx}g\left(x\right)\\ \mathrm{Power}\mathrm{rule}:\frac{d}{dx}{x}^n=n{x}^{n-1}\end{array}\right] \\ =2\sec \left(x\right)\times \sec \left(x\right)\tan \left(x\right) \\ =2{\sec }^2\left(x\right)\tan \left(x\right) \end{gathered}$$

Therefore, the Derivative of $y={\sec }^2\left(x\right)$ is $2{\sec }^2\left(x\right)\tan \left(x\right)$.