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How to Take the Derivative of tan(x)

How to Take the Derivative of tan(x)
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Lesson Transcript
Instructor: Gerald Lemay

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

In this lesson we show how to take the derivative of the tangent function including the cases when the argument is a function of x. As an application, we show how this derivative is used for approximating a function.

How to Take the Derivative of Tangent

To find the derivative of tangent of x, we'll start by writing tan x as sin x/cos x and then use the quotient rule to differentiate. The quotient rule says that if two functions are differentiable, then the quotient is also differentiable. Here's the quotient rule applied to tan x when in form of sin x/cos x:


quotient_rule_derivative_tan_x


Now we know that the derivative of sin x is cos x and the derivative of cos x is -sin x. Substituting these derivatives in the parentheses and simplifying, we get:


simplify


Now there are two trigonometric identities we can use to simplify this problem:

  • sin2x + cos2x = 1
  • sec x = 1/cos x


the_result


And that's it, we are done! The derivative of tan x is sec2x.

However, there may be more to finding derivatives of tangent. In the general case, tan x is the tangent of a function of x, such as tan g(x). Note in the simple case, g(x) = x.

Generally, we are looking for:


general_form_for_derivative_of_tan_g(x)


For example:


g(x)=3x


Let's look at the steps to solve this problem.

The Step-by-Step Approach:

Step 1: Identify g(x)

In this example, g(x) = 3x

Step 2: Find g'(x)

Taking the derivative of g(x), we get:


g_prime_of_x


Step 3: Use the Chain Rule


general_form_of_derivative


So in this example:


3sec^2(3x)


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