Differentiate the following function: tan²x

To differentiate the function \( \tan^2 x \), we will use the chain rule. The chain rule states that if you have a composite function, you can differentiate it by taking the derivative of the outer function and multiplying it by the derivative of the inner function.

Step-by-Step Differentiation

Let’s break it down:

• Let \( u = \tan x \). Then, the function can be rewritten as \( u^2 \).
• Now, differentiate \( u^2 \) with respect to \( u \):

Derivative of \( u^2 \): The derivative is \( 2u \).

• Next, we need to find the derivative of \( u = \tan x \):

Derivative of \( \tan x \): The derivative is \( \sec^2 x \).

Applying the Chain Rule

Now, we combine these results using the chain rule:

The derivative of \( \tan^2 x \) is:

Final Result: \( \frac{d}{dx}(\tan^2 x) = 2\tan x \cdot \sec^2 x \).

So, the derivative of \( \tan^2 x \) is \( 2\tan x \sec^2 x \).