Question

Differentiate $secx$ by the first principle.

Answer 1

Let us consider

$secx=\frac{1}{\cos x}$

By quotient rule derivative of $secx=\frac{1}{\cos x}$

[The quotient rule of differentiation is defined as the ratio of two functions (1st function / 2nd function ), is equal to the ratio of (differentiation of 1st function $\times$the 2nd function $-$ differentiation of the second function $\times$the first function ) to the square of the 2nd function]

so,

$\frac{\frac{d}{dx}\times 1-\frac{d}{dx}\cos x\times 1}{{(\cos x)}^2}$

by simplifying the equation

$$\begin{gathered} =\frac{0\times \cos x-{\displaystyle \frac{d}{dx}}\cos x\times 1}{{\left(\cos x\right)}^2} \\ =\frac{\sin x}{\cos x\times \cos x} \\ =\frac{\sin x}{\cos x}\times \frac{1}{\cos x} \\ =tanx\times secx \end{gathered}$$

Thus, differentiation of $secx$ by the first principle is$\mathbf{}\mathit{t}\mathit{a}\mathit{n}\mathit{x}\mathbf{\times }\mathit{s}\mathit{e}\mathit{c}\mathit{x}$.