Before going to find the derivative that sec x, let united state recall a few things. Sec x is the reciprocal of cos x and tan x is the ratio of sin x and also cos x. These meanings of sec x and tan x are really important to perform the differentiation that sec x with respect come x. We can find it using various ways such as:

by making use of the very first principleby using the quotient ruleby utilizing the chain rule

Let us perform the differentiation that sec x in every of this methods and also we will fix a couple of problems making use of the derivative that sec x.

You are watching: Prove that d/dx(secx)=secx tanx

1.What is Derivative that Sec x?
2.Derivative that Sec x by very first Principle
3.Derivative that Sec x through Quotient Rule
4.Derivative that Sec x by Chain Rule
5.FAQs top top Derivative of Sec x

What is Derivative the Sec x?


The derivative of sec x v respect come x is sec x · tan x. I.e., that is the product of sec x and tan x. We represent the derivative of sec x with respect come x v d/dx(sec x) (or) (sec x)'. Thus,

d/dx (sec x) = sec x · tan x (or)(sec x)' = sec x · tan x

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But wherein is tan x coming from in the derivative the sec x? We space going to differentiate sec x in various methods such as making use of the very first principles (definition of the derivative), quotient rule, and chain preeminence in the upcoming sections.


Derivative the Sec x by first Principle


We are going to prove that the derivative of sec x is sec x · tan x by making use of the first principles (or) the an interpretation of the derivative. For this, assume the f(x) = sec x.

Proof:

By first principle, the derivative the a function f(x) is,

f'(x) = limₕ→₀ / h ... (1)

Since f(x) = sec x, we have f(x + h) = sec (x + h).

Substituting these worths in (1),

f' (x) = limₕ→₀ /h

= limₕ→₀ 1/h <1/(cos (x + h) - 1/cos x)>

= limₕ→₀ 1/h /

By sum to product formulas, cos A - cos B = -2 sin (A+B)/2 sin (A-B)/2. So

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

= 1/cos x limₕ→₀ 1/h <- 2 sin (2x + h)/2 sin (- h)/2> /

Multiply and divide through h/2,

= 1/cos x limₕ→₀ (1/h) (h/2) <- 2 sin (2x + h)/2 sin (- h/2) / (h/2)> /

When h → 0, we have 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)

We have limₓ→₀ (sin x) / x = 1. So

f'(x) = 1/cos x. 1. Sin x/cos x

We recognize that 1/cos x = sec x and also sin x/cos x = tan x. So

f'(x) = sec x · tan x

Hence proved.


Derivative that Sec x by Quotient Rule


We will prove that the differentiation of sec x with respect come x offers sec x · tan x by making use of the quotient rule. Because that this, we will certainly assume the f(x) = sec x and it deserve to be created as f(x) = 1/cos x.

Proof:

We have actually f(x) = 1/cos x = u/v

By quotient rule,

f'(x) = (vu' - uv') / v2

f'(x) = / (cos x)2

= / cos2x

= (sin x) / cos2x

= 1/cos x · (sin x)/(cos x)

= sec x · tan x

Hence proved.


Derivative that Sec x through Chain Rule


To prove that the derivative that sec x to be sec x · tan x by chain rule, we will assume that f(x) = sec x = 1/cos x.

Proof:

We deserve to write f(x) as,

f(x) = 1/cos x = (cos x)-1

By power rule and also chain rule,

f'(x) = (-1) (cos x)-2 d/dx(cos x)

By a residential property of exponents, a-m = 1/am. Also, we recognize that d/dx(cos x) = - sin x. So

f'(x) = -1/cos2x · (- sin x)

= (sin x) / cos2x

= 1/cos x · (sin x)/(cos x)

= sec x · tan x

Hence proved.

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