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The tangent function, denoted as tan(x), is a trigonometric function that relates the ratio of the sine and cosine of an angle. The rate of change or derivative of the tangent function varies depending on the input value (x).

Let's explore the rates for tan(x) in different scenarios:

1. In general, the derivative of the tangent function is given by:

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

Here, sec(x) represents the secant function, which is the reciprocal of the cosine function.

2. At specific points:

At x = 0 degrees (or multiples of 180 degrees):

The tangent function has an undefined rate (or vertical asymptote) at these points. The derivative is not defined.

At x = 45 degrees (or π/4 radians):

The tangent function has a slope of 1 at this point. The derivative is equal to 1.

At x = 90 degrees (or π/2 radians):

The tangent function has an undefined rate (or vertical asymptote) at this point. The derivative is not defined.

At x = 180 degrees (or π radians):

The tangent function has an undefined rate (or vertical asymptote) at this point. The derivative is not defined.

At x = 270 degrees (or 3π/2 radians):

The tangent function has an undefined rate (or vertical asymptote) at this point. The derivative is not defined.

At x = 360 degrees (or 2π radians):

The tangent function has an undefined rate (or vertical asymptote) at this point. The derivative is not defined.

3. Generally, the tangent function has periodic behavior with a period of 180 degrees (or π radians). The derivative of tan(x) repeats this pattern as well.

For such more question on tangent:

https://brainly.com/question/3685164

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