Compute the first derivative of the function.
h(x) = arctan(8x2 + 9)

To find the first derivative of the function \( h(x) = \arctan(8x^2 + 9) \), we'll apply the chain rule, since the function involves the composition of functions.

The derivative of \( \arctan(u) \) with respect to \( u \) is:

\[
\frac{d}{du} \arctan(u) = \frac{1}{1 + u^2}
\]

Let \( u = 8x^2 + 9 \), so now we need to apply the chain rule.

### Steps:
1. Differentiate the outer function (arctangent):
   \[
   \frac{d}{dx} \arctan(8x^2 + 9) = \frac{1}{1 + (8x^2 + 9)^2}
   \]

2. Differentiate the inner function \( 8x^2 + 9 \):
   \[
   \frac{d}{dx} (8x^2 + 9) = 16x
   \]

3. Multiply the derivatives from steps 1 and 2:
   \[
   h'(x) = \frac{1}{1 + (8x^2 + 9)^2} \cdot 16x
   \]

Thus, the derivative is:

\[
h'(x) = \frac{16x}{1 + (8x^2 + 9)^2}
\]

Let me know if you'd like further details or clarification.

Here are 5 related questions to deepen your understanding:
1. What is the derivative of \( \arctan(kx) \) for some constant \( k \)?
2. How does the chain rule apply when differentiating composite functions?
3. What are the critical points of \( h(x) = \arctan(8x^2 + 9) \)?
4. What is the second derivative of \( h(x) \)?
5. How does the graph of \( h(x) = \arctan(8x^2 + 9) \) behave as \( x \to \infty \)?

**Tip:** Always remember that the chain rule is essential when differentiating composite functions, especially when the outer function has its own derivative formula.

Compute the first derivative of the function h(x) = arctan(8x² + 9).

Find the first derivative of h(x) = arctan(8x² + 9) using the chain rule. Learn how to apply trigonometric differentiation and the chain rule to compute this derivative, with step-by-step explanations. Suitable for advanced high school and college students.

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