Derivative of tan x
sec^2 x
The derivative of tan x can be found by using the quotient rule of differentiation. Recall that tan x is defined as sin x / cos x. Therefore, we have:
tan x = sin x / cos x
Now, let’s find the derivative of this expression with respect to x:
(d/dx) [tan x] = (d/dx) [sin x / cos x]
Using the quotient rule, we get:
(d/dx) [tan x] = [(cos x)(d/dx)(sin x) – (sin x)(d/dx)(cos x)] / (cos x)^2
Simplifying this expression using trigonometric identities, we get:
(d/dx) [tan x] = [cos^2(x) – sin^2(x)] / cos^2(x)
Simplifying further, we get:
(d/dx) [tan x] = 1 / cos^2(x)
Recall that cos^2(x) = 1 / [1 + tan^2(x)], therefore:
(d/dx) [tan x] = 1 / [cos^2(x)] = 1 / [1 + tan^2(x)]
Therefore, the derivative of tan x is 1 / [1 + tan^2(x)].
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