Question

$\frac{d}{dx}\sec \left(\mathrm{x}\right)$

Answer 1

Differentiate $\sec \left(x\right)$

$$\begin{gathered} \frac{\mathrm{d}}{dx}\sec \left(x\right) \\ \Rightarrow \frac{\mathrm{d}}{\mathrm{d}x}\frac{1}{\cos \left(\mathrm{x}\right)} \\ =\frac{\cos \left(x\right)·{\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}}\left(1\right)-1·{\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}}\cos \left(x\right)}{\cos {\left(x\right)}^2}\left[\because \frac{\mathbf{d}}{\mathbf{d}\mathbf{x}}\mathbf{\left(}\frac{\mathbf{u}}{\mathbf{v}}\mathbf{\right)}\mathbf{=}\frac{\mathbf{v}{\displaystyle \frac{\mathbf{d}}{\mathbf{d}\mathbf{x}}}\mathbf{u}\mathbf{-}\mathbf{u}{\displaystyle \frac{\mathbf{d}}{\mathbf{d}\mathbf{x}}}\mathbf{v}}{{\mathbf{v}}^{\mathbf{2}}}\right] \\ =\frac{0-\left[-\sin \left(x\right)\right]}{\cos {\left(x\right)}^2} \\ =\frac{\sin \left(x\right)}{\cos \left(x\right)}·\frac{1}{\cos \left(x\right)} \\ =\tan \left(x\right)·\sec \left(x\right) \end{gathered}$$

Therefore, $\frac{d}{dx}\sec \left(\mathrm{x}\right)=\tan \left(x\right)·\sec \left(x\right)$.