# Derivative of sec x: Proof by Quotient, Chain & First Principle

In this article, you will learn what is the derivative of sec x as well as prove the derivative of sec x by quotient rule, first principal rule, and chain rule.

I have already discussed the derivative of sin x, cos x, and tan x in a previous article you can check from here.

So, without wasting time let's get started.

## What is sec x?

Sec x is a trigonometric function that is reciprocal of cos x.

## Derivative of sec x

The derivative of sec x is equal to the product of sec x and tan x.

We can prove the derivative of sec x in three ways first by using the quotient rule and second by using the first principle rule and the last chain rule.

### Derivative of sec x Proof by Quotient Rule

The formula of the quotient rule is,

dy/dx =  {v (du/dx) - u (dv/dx)}/v²

Where,

dy/dx = derivative of y with respect to x

v = variable v

du/dx = derivative of u with respect to x

u = variable u

dv/dx = derivative of v with respect to x

v = variable v

Let us,

y = sec x

As we know,

sec x  = 1/cos x

So,

We can written as,

y  = 1/cos x.

Where,

u = 1

v = cos x

Now putting these values on the quotient rule formula, we will get

dy/dx = [cos x d/dx(1) - 1 d/dx(cos x)] / (cos x)²

= [cos x (0) - 1 (-sin x)] / (cos²x)

= (sin x) / (cos²x)

= (1/cos x) × {(sin x)/(cos x)}

So, as we know,

1/cos x = sec x  and  sin x/cos x = tan x

Hence, after putting these values we will get,

d/dx (sec x) = sec x · tan x

Thus, we proved the derivative of sec x will be equal to sec x · tan x using the quotient rule method.

### Derivative of sec x Proof by First Principle Rule

According to the first principle rule, the derivative limit of a function can be determined by computing the formula:

For a differentiable function y = f (x)

We define its derivative w.r.t  x as :

dy/dx = f ' (x) = limₕ→₀ [f(x+h) - f(x)]/h

f'(x) = limₕ→₀ [f(x+h) - f(x)]/h

This limit is used to represent the instantaneous rate of change of the function f(x).

Let,

f (x) = sec x

So,

f(x + h) = sec (x + h)

Putting these values on the above first principle rules equation.

f' (x) = limₕ→₀ [sec (x + h) - sec x]/h

So, as we know sec x = 1/cos x

Hence, after putting 1/cos x  in place of sec x we will get,

= limₕ→₀ 1/h [1/(cos (x + h) - 1/cos x)]

After solving this we will get,

= limₕ→₀ 1/h [cos x - cos(x + h)] / [cos x cos(x + h)]

So, as we know
cos a - cos b = -2 [{sin (a + b)/ 2} × {sin (a - b)/2}]

Now putting these values,

f'(x) = 1/cos x limₕ→₀ 1/h [- 2 sin (x + x + h)/2 × sin (x - x - h)/2] / [cos(x + h)]

= 1/cos x limₕ→₀ 1/h [- 2 sin (2x + h)/2 sin (- h)/2] / [cos(x + h)]

Now multiply and divide by h/2 we will get,

= 1/cos x limₕ→₀ (1/h) (h/2) [- 2 sin (2x + h)/2 sin (- h/2) / (h/2)] / [cos(x + h)]

If  h→₀ then also h/2→₀

So

f'(x) = 1/cos x limₕ/₂→₀ sin (h/2) / (h/2) × limₕ→₀ (sin(2x + h)/2)/cos(x + h)

So, for limₓ→₀ sin x / x = 1

Hence,

f'(x) = 1/cos x × 1 × sin x/cos x

As we know,

1/cos x = sec x

sin x/cos x = tan x

After putting these values we will get,

f'(x) = sec x · tan x

Thus, we proved the derivative of sec x will be equal to sec x · tan x using the first principle rule method.

### Derivative of sec x Proof by Chain Rule

Let us,

y = sec x

As we know,

sec x = 1/cos x

So,

y = 1 / (cos x)

= (cos x)⁻¹

By using the chain rule,

The formula of chain rule is,

dy/dx = (dy/du) × (du/dx)

Where,

dy/dx = derivative of y with respect to x

dy/du = derivative of y with respect to u

du/dx = derivative of u with respect to x

After putting these values we can find,

dy/dx = (-1) (cos x)⁻² d/dx(cos x)

Since,

d/dx(cos x) = - sin x and a⁻ⁿ = 1/aⁿ

Hence,

dy/dx  = -1/cos²x · (- sin x)

= (sin x) / cos²x

= 1/cos x · (sin x)/(cos x)

d/dx (sec x) = sec x · tan x

Thus, we proved the derivative of sec x will be equal to sec x · tan x using the chain rule method.

## FAQ Related to the Derivative of sec x

### What is the derivative of sec(x)?

The derivative of sec x is equal to the product of sec x and tan x.

### Is secant the inverse of cosine?

Yes, secant is the inverse of cosine.

### What is the derivative of the sec inverse of x?

The derivative of the sec inverse of x is the 1/(x √x² - 1).

So friends here I discussed all aspects related to the derivative of sec x

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