Learn To Find The Derivative Of Tangent Function Using Quotient Rule With Step-By-Step Guide

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).

More Answers: