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How to differentiate tan x

How to differentiate tan x

31.01.2023

Calculus can sometimes feel intimidating, but breaking down problems into manageable steps makes the process much easier. In this tutorial, we'll explore how to differentiate the tangent function, tan x, using two common methods: the quotient rule and trigonometric identities.

Understanding the Tangent Function

The tangent function is defined as the ratio of sine to cosine:

Tangent Function

This expression will be very useful when we apply differentiation techniques.

Method 1: Using the Quotient Rule

The quotient rule is a formula for differentiating a function that is the quotient of two differentiable functions. Recall that if

Using the Quotient Rule step 1

then the derivative f'(x) is given by:

Using the Quotient Rule step 2

For tan x:

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

Plugging into the quotient rule, we have:

Using the Quotient Rule step 3

Simplify the numerator:

Simplify the numerator

Recall the Pythagorean identity:

Recall the Pythagorean identity

Thus, the derivative simplifies to:

derivative simplifies to

This is often written as:

final step

Method 2: Using Trigonometric Identities

Another approach involves differentiating the tangent function directly, knowing that

Using Trigonometric Identities

While this result is usually provided as a standard derivative in calculus textbooks, understanding the derivation reinforces the concept. We can derive it by starting from the definition tan x = sin x/cos x and applying the quotient rule (as shown above). This method highlights the beauty of interconnected trigonometric identities.

Summary

To differentiate tan x, we:

  1. Expressed tan x as sin x/cos x.
  2. Applied the quotient rule with sin x and cos x.
  3. Simplified the expression using the Pythagorean identity to obtain sec^2 x.

Thus, we conclude:

differentiate tan x

Final Thoughts

Understanding how to differentiate tan x is not only a critical calculus skill but also a gateway to more advanced topics in mathematics. By mastering both the quotient rule and the related trigonometric identities, you'll be well-equipped to tackle a wide range of differentiation problems.

Feel free to leave a comment below if you have any questions or need further clarification on any of the steps. Happy differentiating!

Author: MyEdSpace
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