Derivative of Sec x
Before going to find the derivative of sec x, let us recall a few things. sec x is the reciprocal of cos x and tan x is the ratio of sin x and cos x. These definitions of sec x and tan x are very important to do the differentiation of sec x with respect to x. We can find it using various ways such as:
- by using the first principle
- by using the quotient rule
- by using the chain rule
Let us do the differentiation of sec x in each of these methods and we will solve a few problems using the derivative of sec x.
| 1. | What is Derivative of Sec x? |
| 2. | Derivative of Sec x by First Principle |
| 3. | Derivative of Sec x by Quotient Rule |
| 4. | Derivative of Sec x by Chain Rule |
| 5. | FAQs on Derivative of Sec x |
What is Derivative of Sec x?
The derivative of sec x with respect to x is sec x · tan x. i.e., it is the product of sec x and tan x. We denote the derivative of sec x with respect to x with d/dx(sec x) (or) (sec x)'. Thus,
- d/dx (sec x) = sec x · tan x (or)
- (sec x)' = sec x · tan x
But where is tan x coming from in the derivative of sec x? We are going to differentiate sec x in various methods such as using the first principles (definition of the derivative), quotient rule, and chain rule in the upcoming sections.
Derivative of Sec x by First Principle
We are going to prove that the derivative of sec x is sec x · tan x by using the first principles (or) the definition of the derivative. For this, assume that f(x) = sec x.
Proof:
By first principle, the derivative of a function f(x) is,
f'(x) = limₕ→₀ [f(x + h) - f(x)] / h ... (1)
Since f(x) = sec x, we have f(x + h) = sec (x + h).
Substituting these values in (1),
f' (x) = limₕ→₀ [sec (x + h) - sec x]/h
= limₕ→₀ 1/h [1/(cos (x + h) - 1/cos x)]
= limₕ→₀ 1/h [cos x - cos(x + h)] / [cos x cos(x + 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] / [cos(x + h)]
= 1/cos x limₕ→₀ 1/h [- 2 sin (2x + h)/2 sin (- h)/2] / [cos(x + h)]
Multiply and divide by h/2,
= 1/cos x limₕ→₀ (1/h) (h/2) [- 2 sin (2x + h)/2 sin (- h/2) / (h/2)] / [cos(x + h)]
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 know that 1/cos x = sec x and sin x/cos x = tan x. So
f'(x) = sec x · tan x
Hence proved.
Derivative of Sec x by Quotient Rule
We will prove that the differentiation of sec x with respect to x gives sec x · tan x by using the quotient rule. For this, we will assume that f(x) = sec x and it can be written as f(x) = 1/cos x.
Proof:
We have f(x) = 1/cos x = u/v
By quotient rule,
f'(x) = (vu' - uv') / v2
f'(x) = [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)
= sec x · tan x
Hence proved.
Derivative of Sec x by Chain Rule
To prove that the derivative of sec x to be sec x · tan x by chain rule, we will assume that f(x) = sec x = 1/cos x.
Proof:
We can write f(x) as,
f(x) = 1/cos x = (cos x)-1
By power rule and chain rule,
f'(x) = (-1) (cos x)-2 d/dx(cos x)
By a property of exponents, a-m = 1/am. Also, we know 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.
Topics Related to Derivative of Sec x:
Here are some topics that you may be interested in while studying about the differentiation of sec x.
FAQs on Derivative of Sec x
What is Derivative of Sec x With Respect to x?
The derivative of sec x with respect to x is written as d/dx(sec x) and it is equal to sec x tan x. i.e., the differentiation of sec x is the product of sec x and tan x.
How to Find Derivative of Sec x by First Principle?
The first principle is used to find the derivative of a function f(x) using the formula f'(x) = limₕ→₀ [f(x + h) - f(x)] / h. By substituting f(x) = sec x and f(x + h) = sec (x + h) in this formula and simplifying it, we can find the derivative of sec x to be sec x tan x. For more detailed proof, click here.
What is the Differentiation of Sec Square x?
Sec square x can be written as f(x) = (sec x)2. By power rule and chain rule, its derivative is, f'(x) = 2 sec x d/dx(sec x) = 2 sec x (sec x tan x) = 2 sec2x tan x.
Is the Derivative of Sec Inverse x Same as the Derivative of Sec x?
No, the derivative of sec x is NOT same as the derivative of sec-1x. The derivative of sec x is sec x tan x whereas the derivative of sec-1x is 1/(x √x² - 1).
What is the Derivative of Sec x²?
We know that the derivative of sec w is sec w tan w. Also, by using the chain rule, d/dx (sec x²) = sec x² tan x² d/dx(x²) = 2x sec x² tan x².
What is the Derivative of Sec x with Respect to Tan x?
We have to find d(sec x) / d(tan x). Let sec x = u and tan x = v. Then we have to find du/dv. Now, du/dx = sec x tan x and dv/dx = sec²x. Now,
du/dv = (du/dx) / (dv/dx)
= (sec x tan x) / (sec²x)
= (tan x) / (sec x)
= [sin x/ cos x] / [1/cos x]
= sin x.
Thus, the derivative of sec x with respect to tan is sin x.
How To Prove that the Differentiation of Sec x is Sec x Tan x?
We can prove that the derivative of sec x is sec x tan x using different methods. For detailed information, you can click on one of the following: