What is the derivative of tangent of x (tan x)?

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The derivative of the tangent function, denoted as tan(x), is a fundamental concept in calculus, especially for those working in fields that involve trigonometric functions.

The derivative of tan(x) can be derived from the definition of the tangent function itself:

  • tan(x) = sin(x) / cos(x)

To find the derivative, we can apply the quotient rule. The quotient rule states that if you have a function that is the division of two other functions f(x) = g(x)/h(x), then the derivative is given by:

f'(x) = (g'(x)h(x) - g(x)h'(x)) / [h(x)]^2

In our case:

  • g(x) = sin(x) with g'(x) = cos(x)
  • h(x) = cos(x) with h'(x) = -sin(x)

Plugging these into the quotient rule gives:

tan'(x) = (cos(x) imes cos(x) - sin(x) imes (-sin(x))) / [cos(x)]^2

Upon simplifying this, we get:

tan'(x) = (cos2(x) + sin2(x)) / [cos(x)]^2

Next, we apply the Pythagorean identity sin2(x) + cos2(x) = 1. This leads us to:

tan'(x) = 1 / [cos(x)]^2

Or, using a different notation:

tan'(x) = sec2(x)

Thus, the derivative of tan(x) is:

sec2(x)

In summary, the derivative of the tangent function is one of the essential derivatives to memorize, especially if you plan to delve deeper into calculus or related fields. Understanding this derivative is crucial as it assists not only in solving problems but also in effectively working with integrals and other derivatives involving trigonometric functions.

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