# Differentiation of tan x

#### Understand the Problem

The question is asking for the derivative of the tangent function with respect to x, which involves applying the rules of differentiation in calculus.

The derivative of the tangent function is $f'(x) = \sec^2(x)$.

The derivative of the tangent function is $f'(x) = \sec^2(x)$.

#### Steps to Solve

1. Identify the function to differentiate

The function we want to differentiate is the tangent function, which is written as $f(x) = \tan(x)$.

1. Use the derivative rule for tangent

The derivative of the tangent function is found using the formula: $$\frac{d}{dx}(\tan(x)) = \sec^2(x)$$

1. Apply the derivative rule

Since our function is $f(x) = \tan(x)$, we apply the derivative rule: $$f'(x) = \sec^2(x)$$

The derivative of the tangent function is $f'(x) = \sec^2(x)$.

The derivative of the tangent function is a fundamental result in calculus. It shows how the slope of the tangent line to the curve changes at any point on the function. An interesting fact is that the secant function, $\sec(x)$, is itself related to the cosine function, as it is defined as the reciprocal of cosine, $\sec(x) = \frac{1}{\cos(x)}$.