Defreantiation of combosit function tan x

To differentiate the composite function \( \tan(x) \), we use the chain rule. 

Given \( f(x) = \tan(x) \), we know:

\[
\frac{d}{dx} [\tan(x)] = \sec^2(x)
\]

The chain rule states that if you have a composite function \( f(g(x)) \), the derivative is:

\[
\frac{d}{dx} [f(g(x))] = f'(g(x)) \cdot g'(x)
\]

In this case, since the inner function is just \( x \), and its derivative is \( 1 \), the derivative of \( \tan(x) \) is simply:

\[
\frac{d}{dx} [\tan(x)] = \sec^2(x) \cdot 1 = \sec^2(x)
\]

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

Would you like more details or have any questions about this? Here are some related questions you might find interesting:

1. What is the derivative of \( \sec(x) \)?
2. How does the chain rule work with more complex composite functions?
3. Can you derive \( \sec^2(x) \) from the identity \( 1 + \tan^2(x) = \sec^2(x) \)?
4. What is the derivative of \( \tan^{-1}(x) \)?
5. How do you differentiate \( \tan(2x) \)?
6. What is the second derivative of \( \tan(x) \)?
7. How does the product rule apply to trigonometric functions?
8. What are some applications of differentiating \( \tan(x) \)?

**Tip:** Always check your trigonometric identities before differentiating, as they can simplify your work significantly.

Learn how to differentiate the composite function \( \tan(x) \) using the chain rule and trigonometric identities. Detailed step-by-step solution provided.

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