How to differentiate tan x?

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:

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

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

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:

Simplify the numerator:

Recall the Pythagorean identity:

Thus, the derivative simplifies to:

This is often written as:

Method 2: Using trigonometric identities

Another approach involves differentiating the tangent function directly, knowing that

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.

How to differentiate tan: A 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:

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