To find the derivative of \( \tan(\sqrt{x}) \), we can use the **chain rule**.

Let \( y = \tan(\sqrt{x}) \). The derivative of \( y \) with respect to \( x \) is:

1. First, differentiate the outer function \( \tan(u) \) with respect to \( u \), where \( u = \sqrt{x} \), using the fact that \( \frac{d}{du} \tan(u) = \sec^2(u) \).
   
   So, \( \frac{d}{du} \tan(\sqrt{x}) = \sec^2(\sqrt{x}) \).

2. Next, differentiate the inner function \( \sqrt{x} \) with respect to \( x \). We know that \( \frac{d}{dx} \sqrt{x} = \frac{1}{2\sqrt{x}} \).

3. Now, apply the chain rule:

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

Thus, the derivative of \( \tan(\sqrt{x}) \) is:

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

Let me know if you'd like a more detailed explanation or if you have any questions!

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Here are five related questions you might want to explore:
1. What is the derivative of \( \tan^2(x) \)?
2. How do you find the derivative of \( \sec(\sqrt{x}) \)?
3. What is the derivative of \( \tan^3(\sqrt{x}) \)?
4. Can you find the second derivative of \( \tan(\sqrt{x}) \)?
5. How do you compute the derivative of \( \tan(x^2) \)?

**Tip:** When differentiating composite functions, always apply the chain rule carefully by breaking the problem into outer and inner functions.

Learn how to differentiate the trigonometric function tan(sqrt(x)) using the chain rule. This solution involves key calculus concepts like the derivative of trigonometric functions and the square root function. Suitable for students in Grades 11-12 and college-level calculus courses.

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