Derivative of tan x
The derivative of tan x can be found using the quotient rule
The derivative of tan x can be found using the quotient rule. Let’s go step by step.
The quotient rule states that for two functions u(x) and v(x), the derivative of their quotient u(x)/v(x) is given by:
(u'(x)v(x) – u(x)v'(x)) / (v(x))^2
In this case, u(x) = sin x and v(x) = cos x.
So, applying the quotient rule, we have:
[tan x]’ = (sin’ x * cos x – sin x * cos’ x) / (cos x)^2
We know that the derivatives of sin x and cos x are cos x and -sin x, respectively. Plugging these values into the equation, we get:
[tan x]’ = (cos x * cos x – sin x * (-sin x)) / (cos x)^2
Simplifying further, we have:
[tan x]’ = (cos^2 x + sin^2 x) / (cos x)^2
Using the Pythagorean identity sin^2 x + cos^2 x = 1, the equation becomes:
[tan x]’ = 1 / (cos x)^2
Another way to represent this is:
[tan x]’ = sec^2 x
Therefore, the derivative of tan x is sec^2 x.
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