Calculate the derivative of tan(x).

The derivative of tan(x) is sec^2(x).

To find the derivative of tan(x), we can use the quotient rule. Recall that tan(x) = sin(x)/cos(x). Therefore, we have:

tan'(x) = [cos(x)(sin'(x)) - sin(x)(cos'(x))] / cos^2(x)

Using the derivatives of sin(x) and cos(x), we get:

tan'(x) = [cos(x)cos(x) - sin(x)(-sin(x))] / cos^2(x)

tan'(x) = [cos^2(x) + sin^2(x)] / cos^2(x)

Using the identity cos^2(x) + sin^2(x) = 1, we get:

tan'(x) = 1 / cos^2(x)

Recall that sec(x) = 1 / cos(x). Therefore, we have:

tan'(x) = sec^2(x)

This is the final answer. We have shown that the derivative of tan(x) is sec^2(x).