The derivative of sec x with respect to x is equal to secx tanx. The secx is the reciprocal of cosx. In this post, we will learn how to find the derivatives of sec x using the following methods:

- First principle of derivatives
- Chain rule of derivatives
- Quotient rule of derivatives.

Table of Contents

## What is the Derivative of Sec x?

The derivative of secx with respect to x is denoted by the symbol $\frac{d}{dx}$(sec x) or (sec x)$’$ and it is equal to secx tanx. Using the fact $\sec x =\frac{1}{\cos x}$, we can find the derivative of sec x by the chain rule and quotient rule of derivatives.

## Derivative of Sec x Formula

The formula for the derivative of secx is given below.

- $\frac{d}{dx}$(sec x) = sec x tan x, (or)
- (sec x)$’$ = sec x tan x.

## Derivative of Sec x by First Principle

We will show that the differentiation of sec x is sec x tan x by the first principle. Putting f(x) = sec x in the derivative formula by first principle $f'(x)$ $=\lim\limits_{h \to 0} \frac{f(x+h)-f(x)}{h}$, we get the derivative of sec x. Therefore,

## Derivative of Sec x by Chain Rule

By the chain rule, we will now show that the differentiation of sec x is sec x tan x. We have

$\sec x =\dfrac{1}{\cos x}$

Put t=cosx. So $\dfrac{dt}{dx}=-\sin x$. Now,

$\dfrac{d}{dx}(\sec x)=\dfrac{d}{dx}(\dfrac{1}{\cos x})$

= $\dfrac{d}{dt}(\dfrac{1}{t}) \cdot \dfrac{dt}{dx}$, by the chain rule of derivatives

= $\dfrac{d}{dt}$(t^{-1}) ⋅ (-sinx)

= -t^{-1-1} ⋅ (-sinx) by the power rule of derivatives

= $\dfrac{\sin x}{\cos^2 x}$ as t=cosx.

= secx tanx.

So the derivative of secx by chain rule is secx tanx.

**Also Read:**

Derivative of tanx | The derivative of tanx is sec^{2}x. |

Derivative of cotx | The derivative of cotx is -cosec^{2}x. |

Derivative of sin^{2}x | The derivative of sin^{2}x is 2sinx cosx. |

Derivative of e^{cosx} | The derivative of e^{cosx} is -sinx e^{cosx}. |

## Derivative of Sec x by Quotient Rule

We will now find the derivative of secx by the quotient rule. This rule is used to find the derivative of a quotient function which says that

$(\dfrac{f}{g})’=\dfrac{gf’-fg’}{g^2}$, where $’$ denotes the first order derivative.

Let us express secx as a quotient function as follows.

$\sec x =\dfrac{1}{\cos x}$

Applying the above quotient rule of derivatives with f=1 and g=cosx, we get that

$(\sec x)’ =\dfrac{\cos x \cdot 1^\prime – 1 \cdot (\cos x)’}{\cos^2 x}$

= $\dfrac{\cos x \cdot 0 – 1 \cdot (-\sin x)}{\cos^2 x}$

= $\dfrac{\sin x}{\cos^2 x}$ = secx tanx.

So the derivative of secx by quotient rule is secx tanx.

## FAQs on Derivative of Sec x

**Q1: what is the derivative of secx?**

Answer: The derivative of secx is secx tanx. This can proved using the first principle, chain rule, and quotient rule of derivatives.

**Q2: Is secx differentiable at the point x=0?**

Answer: As the derivative of secx is secx tanx = sinx/cos^{2}x and cosx is undefined at x=0, the function secx is not differentiable at x=0.

**Q3: What is the integration of secx?**

Answer: The integration of secx is ln|secx+tanx|+C where C is an integration constant and ln is the logarithm with base e, called the natural logarithm.

This article is written by Dr. T. Mandal, Ph.D in Mathematics. On Mathstoon.com you will find Maths from very basic level to advanced level. Thanks for visiting.