Question

Find the derivatives of each of the functions in Exercises 17–50. In some cases it may be convenient to do some preliminary algebra.

$f\left(x\right)={\sec }^2\left(x^{-1}\right).$

Short answer

$Thevalueofthederivativeis:-\frac{2}{x^2}se{c}^2\left(\frac{1}{x}\right)\tan \left(\frac{1}{x}\right).$

Step-by-step solution

Step 1. Given Information.

$f\left(x\right)={\sec }^2\left(x^{-1}\right).$

Step 2. General formulas for finding derivatives. 

$$\begin{gathered} \frac{d}{dx}\left(x^n\right)=n{x}^{n-1}, \\ \frac{d}{dx}\left(\sin x\right)=\cos x, \\ \frac{d}{dx}\left(\cos x\right)=-\sin x, \\ \frac{d}{dx}\left(\tan x\right)=se{c}^{2}x, \\ \frac{d}{dx}\left(secx\right)=secx\tan x, \\ \frac{d}{dx}\left(cscx\right)=-cscxcotx, \\ \frac{d}{dx}\left(cotx\right)=-cs{c}^{2}x, \\ Productrule:\frac{d}{dx}\left(uv\right)=u\frac{dv}{dx}+v\frac{du}{dx}, \\ Quotientrule:\frac{d}{dx}\left(\frac{u}{v}\right)=\frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}, \\ Chainrule:\frac{d}{dx}\left(fog\left(x\right)\right)=\frac{d}{dx}\left(f\left(g\right(x\left)\right)\right)\times \frac{d}{dx}\left(g\right(x). \end{gathered}$$

Step 3. Solving derivative using above formulas.

$$\begin{gathered} f\left(x\right)=se{c}^2\left(x^{-1}\right)=se{c}^2\left(\frac{1}{x}\right) \\ Differentiatingu\sin gchainruleweget, \\ \frac{d}{dx}\left(f\right(x\left)\right)=2sec\left(\frac{1}{x}\right)\frac{d}{dx}\left(sec\left(\frac{1}{x}\right)\right) \\ =2sec\left(\frac{1}{x}\right)sec\left(\frac{1}{x}\right)\tan \left(\frac{1}{x}\right)\frac{d}{dx}\left(\frac{1}{x}\right) \\ =2sec\left(\frac{1}{x}\right)sec\left(\frac{1}{x}\right)\tan \left(\frac{1}{x}\right)\left(\frac{-1}{x^2}\right) \\ =-\left(\frac{2}{x^2}\right)se{c}^2\left(\frac{1}{x}\right)\tan \left(\frac{1}{x}\right). \end{gathered}$$