The derivative of the tangent function (tan(x)) is the secant squared function (sec^2(x)). In mathematical notation, it can be represented as:

d/dx(tan(x)) = sec^2(x)

This means that if you have a function y = tan(x), the rate at which y changes with respect to x (its derivative) is given by sec^2(x).

In simpler terms, when you differentiate the tangent function, you get the secant squared function. It tells you how the slope (rate of change) of the tan curve changes as you move along the x-axis. The derivative, sec^2(x), represents the rate of change of the tan function at any given point.

Here are four examples that demonstrate the application of the derivative of the tangent function:

## Example 1: Consider the function y = tan(x).

To find the derivative, we differentiate the tangent function with respect to x:

dy/dx = d/dx(tan(x)) = sec^2(x)

So, the derivative of y = tan(x) is dy/dx = sec^2(x).

## Example 2: Let’s find the derivative of y = 2tan(x).

Again, we differentiate the tangent function and multiply by the constant factor:

dy/dx = 2 * d/dx(tan(x)) = 2 * sec^2(x)

Therefore, the derivative of y = 2tan(x) is dy/dx = 2sec^2(x).

## Example 3: Consider the function y = tan(2x).

To find the derivative, we differentiate the tangent function and apply the chain rule:

dy/dx = d/dx(tan(2x)) = sec^2(2x) * 2

Thus, the derivative of y = tan(2x) is dy/dx = 2sec^2(2x).

## Example 4: Let’s find the derivative of y = tan(x^2).

In this case, we use the chain rule. The derivative of the outer function (tan(u)) is sec^2(u), and the derivative of the inner function (x^2) is 2x. Applying the chain rule, we have:

dy/dx = d/dx(tan(x^2)) = sec^2(x^2) * 2x

Therefore, the derivative of y = tan(x^2) is dy/dx = 2x * sec^2(x^2).

In each example, we differentiate the tangent function with respect to x, and the resulting derivative is the secant squared function. We can apply additional factors or the chain rule when necessary. The derivative gives us information about the rate of change or slope of the tangent function at different points.