Answer: The trigonometric function sec x is the reciprocal of cos x. The derivative of the sine function is expressed as sin′(a)= cos(a), which indicates that the cosine of an angle is what determines the rate of change of sin x at a particular angle, x= an.
Formula for Derivative of Sec x
The derivative or differentiation of a secant function with respect to a variable is equal to the product of the tangent and secant functions. Tan x is the formula for the derivative, and it states that the derivative of the sec x function with respect to x is equal to the product of sec x.
The secant function is expressed in mathematical notation using trigonometry, assuming that x is a variable.
d/dx(secx) = secxtanx
In mathematics, the differentiation of a sec x function for x can be written as (sec x)’ or as d/dx (sec x).
However, you can express the derivative of the secant function in terms of any variable such as:
- d/dh (sec h) = sec h tan h
- d/dw (sec w) = sec w tan w
- d/dy (sec y) = sec y tan y
Proof of the Sec x Derivative
For a given variable, the derivative of a secant function is equal to the product of the tangent and secant functions. The mathematical expression of the secant function, when ‘x’ is utilized as a variable, is x’. The differentiation between “sec x” and “x” is the product of “sec x” and “tan x.” The first principle of differentiation in differential calculus is used to get the derivative of a secant function.
Derivative of Sec x
Let’s review a few things before attempting to get the derivative of sec x. The ratio of sin x to cos x is known as tan x, and the reciprocal of cos x is known as sec x. It is crucial to differentiate sec x from x using sec x and tan x definitions. There are several ways we can find it, including:
- utilizing the first principle
- by applying the rule of quotient
- use the chain rule
Let’s differentiate sec x using each of these techniques, and then use the derivative of sec x to answer a few problems.
Derivative of Sec x by First Principle
Using fundamental principles (or the definition of the derivative), we will demonstrate that sec x’s derivative is sec x · tan x. Now, let’s assume assume f(x) = sec x
Proof: 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)
When these values are substituted 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)]
For cos A – cos B, the sum to product formulas provides cos (A+B)/2 sin (A-B)/2 = -2.
The formula for f’(x) is 1/cos x limₕ→₀ 1/h [- 2 sin (x + x + h)/2 sin (x – x – h)/2]. / [cos (h + x)]
= 1/cos x limₕ→₀ 1/h [- 2 sin (2x + h)/2 sin (- h)/2] / [cos (x + h)]
Divide and multiply 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)]
Sin (h/2) / (h/2) = 1/cos x lim₎/₂→₀ f’(x). cos (x + h)/sin (2x + h) = lim₎→₀
Given limₓ→₀ (sin x) / x = 1, we get,
sin x/cos x = f’(x) = 1/cos x
We are aware that sin x/cos x = tan x and 1/cos x = sec x. Thus,
Sec x · tan x = f’(x)
Sec x Derivative using Quotient Rule
Using the quotient rule, we will demonstrate that sec x · tan x is the result of differentiating sec x about x. To do this, we will suppose that f(x) = sec x, or f(x) = 1/cos x.
Proof: The formula is f(x) = 1/cos x = u/v.
By using the quotient rule,
If vu’ – uv’ = f’(x) / v2, then
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
Chain Rule Derivative of Sec x
Since f(x) = sec x = 1/cos x, we will determine that the derivative of sec x is sec x · tan x by chain rule.
Proof: f(x) can be expressed as,
f(x) = 1/cos x = (cos x)-1
By the rules of chain and power,
f’(x) = (-1) (cos x)-2 d/dx (cos x)
This means that a-m = 1/am by an exponent characteristic. We are also aware that -sin x = d/dx (cos x). So,
f’(x) = -1/cos2x · (- sin x)
(sin x) / cos2x
1/cos x · (sin x)/ (cos x)
sec x · tan x
Solved Examples of Derivative of sec x
Example 1: What is the derivative of (secx)2?
Solution: f(x)=(secx)2
Using the chain rule and the power rule
f′(x)=2secxddx(secx)
=2secx. (secx.tanx)
=2sec2xtanx
Here is the function’s derivative: =2sec2xtanx.
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