Differentiate Tan(x)

The derivative of the tangent function is given by the following formula:

d/dx [tan(x)] = sec²(x)

This can be derived using the basic rules of differentiation and the definition of the tangent function.

The tangent function is defined as the ratio of the sine of an angle to the cosine of that angle. 

In other words, we can write:

tan(x) = sin(x) / cos(x)

We can use the quotient rule of differentiation to find the derivative of this function. The quotient rule states that the derivative of a quotient f(x) / g(x) is given by the following formula:

d/dx [f(x) / g(x)] = [g(x) * d/dx[f(x)] - f(x) * d/dx[g(x)]] / [g(x)]²

Applying this formula to the tangent function, we get:

d/dx [tan(x)] = [cos(x) * d/dx[sin(x)] - sin(x) * d/dx[cos(x)]] / [cos(x)]²

We can use the basic rules of differentiation to evaluate the derivatives on the right-hand side of this equation. The derivative of sin(x) is equal to cos(x), and the derivative of cos(x) is equal to -sin(x). 

Therefore, we can rewrite the above equation as follows:

d/dx [tan(x)] = [cos(x) * cos(x) - sin(x) * (-sin(x))] / [cos(x)]²

Simplifying this further, we get:

d/dx [tan(x)] = [cos²(x) + sin²(x)] / [cos(x)]²

Recall that the sum of the squares of the sine and cosine of an angle is equal to 1 (this is known as the Pythagorean identity). Therefore, we can write:

d/dx [tan(x)] = [1] / [cos(x)]²

The reciprocal of cos²(x) is equal to the square of the reciprocal of cos(x), which is sec²(x). Therefore, we can rewrite the above equation as follows:

d/dx [tan(x)] = sec²(x)

This is the final result. Therefore, the derivative of the tangent function is equal to sec²x).