Derivative of Sec x Examples
Example 1: Find the derivative of sec x ·tan x.
Solution:
Let f(x) = sec x · tan x = u.v
By product rule,
f'(x) = u.v’ + v.u’
⇒ (sec x) d/dx (tan x) + (tan x) d/dx (sec x)
⇒ (sec x)(sec2x) + (tan x) (sec x · tan x)
⇒ sec3x + sec x tan2x
Therefore f'(x)=sec3x + sec x tan2x.
Example 2: Find the derivative of (sec x)2.
Solution:
Let f(x) = (sec x)2
By power rule and chain rule,
f'(x) = 2 sec x d/dx (sec x)
⇒ 2 sec x · (sec x · tan x)
⇒ 2 sec2x tan x
Therefore f'(x)=2 sec2x tan x.
Example 3: Find the derivative of sec-1x.
Solution:
Let y = sec-1x.
Then, sec y = x … (1)
Differentiating both sides with respect to x,
⇒ sec y · tan y (dy/dx) = 1
⇒ dy/dx = 1 / (sec y · tan y)… (2)
By one of the trigonometric identities,
[ tan y = √sec²y – 1 = √x² – 1 ]
⇒ dy/dx = 1/(x √x² – 1)
Therefore f'(x)= 1/(x √x² – 1).
Derivative of Sec x
Derivative of Sec x is sec x tan x. Derivative of Sec x refers to the process of finding the change in the secant function with respect to the independent variable. The specific process of finding the derivative for trigonometric functions is referred to as trigonometric differentiation, and the derivative of Sec x is one of the key results in trigonometric differentiation.
In this article, we will learn about the derivative of sec x and its formula including the proof of the formula using the first principle of derivatives, quotient rule, and chain rule as well.