Derivative of tan x
sec^2 x
The derivative of the tangent function can be found using the quotient rule. Recall that the tangent function is defined as the ratio of the sine and cosine functions:
tan(x) = sin(x)/cos(x)
Using the quotient rule, we have:
d/dx [tan(x)] = [cos(x) * d/dx[sin(x)] – sin(x) * d/dx[cos(x)]] / [cos(x)]^2
Now, we need to find the derivatives of the sine and cosine functions. Recall that the derivative of sin(x) is cos(x) and the derivative of cos(x) is -sin(x):
d/dx [sin(x)] = cos(x)
d/dx [cos(x)] = -sin(x)
Substituting these derivatives back into the quotient rule equation, we get:
d/dx [tan(x)] = [cos(x) * cos(x) – sin(x) * (-sin(x))] / [cos(x)]^2
= [cos^2(x) + sin^2(x)] / [cos(x)]^2
= 1/[cos(x)]^2
Recall that the Pythagorean identity states that sin^2(x) + cos^2(x) = 1. Therefore, we can rewrite the derivative of tangent as:
d/dx [tan(x)] = 1/[cos(x)]^2 = sec^2(x)
So the derivative of tan(x) is sec^2(x).
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