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.

$f\left(x\right)=\sin \left(\cos \right(sec\left(x\right)\left)\right).$

Short answer

The derivative of the given function is:$-\cos \left(\cos \right(sec\left(x\right)\left)\right)\times \sin \left(secx\right)\times \left(secx\tan x\right).$

Step-by-step solution

Step 1. Given Information.

$f\left(x\right)=\sin \left(\cos \right(sec\left(x\right)\left)\right).$

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.

$$\begin{gathered} f\left(x\right)=\sin \left(\cos \right(sec\left(x\right)\left)\right) \\ Differentiatingu\sin gthechainruleweget, \\ \frac{d}{dx}\left(f\right(x\left)\right)=\cos \left(\cos \right(secx\left)\right)\frac{d}{dx}\left(\cos \left(secx\right)\right) \\ =-\cos \left(\cos \right(sec\left(x\right)\left)\right)\times \sin \left(secx\right)\frac{d}{dx}\left(secx\right) \\ =-\cos \left(\cos \right(sec\left(x\right)\left)\right)\times \sin \left(secx\right)\times \left(secx\tan x\right). \end{gathered}$$