The derivative of the tangent function is not sec^2(x). It is actually sec^2(x), which is the square of the secant function.

To explain it to a 5-year-old, imagine you are on a seesaw. When you move up and down on the seesaw, your height changes, and the derivative tells you how fast your height is changing at any point.

Similarly, the tangent function tells us the ratio of the height of a point on a circle to its distance from the center. But we want to know how fast the height is changing as we move around the circle.

To find that out, we use the derivative. The derivative of the tangent function is like a special rule that tells us how fast the height is changing as we move around the circle.

When we differentiate the tangent function, we get a new function called the derivative, which is denoted as d/dx(tan(x)) or tan'(x). And for the tangent function, the derivative is equal to sec^2(x).

So, if we want to know how fast the height is changing at a particular point on the circle, we just put that point into the sec^2(x) formula.

Remember, this is like being on a seesaw. The tangent function tells us the height at different points, and the derivative (sec^2(x)) tells us how fast the height is changing.

Here are three examples demonstrating the calculation of the derivative of the tangent function:

Example 1: Let’s find the derivative of y = tan(x).

To find the derivative, we differentiate the tangent function, which gives us: dy/dx = sec^2(x)

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

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

To find the derivative, we differentiate the tangent function and multiply by the constant factor: dy/dx = 2 * sec^2(x)

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

Example 3: Let’s find the derivative of y = tan(2x).

To find the derivative, we differentiate the tangent function and apply the chain rule: dy/dx = sec^2(2x) * 2

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

In each example, we start with the tangent function and apply the derivative rule to find the corresponding derivative. The derivative of the tangent function is represented by sec^2(x), and we can apply additional factors or chain rule as necessary in more complex examples.