Find the derivative of the function.
h(x) = x2 arctan(7x)

To find the derivative of the function \( h(x) = x^2 \arctan(7x) \), we will use the **product rule** and **chain rule**.

### Steps:
Given:
\[
h(x) = x^2 \arctan(7x)
\]

Let \( u(x) = x^2 \) and \( v(x) = \arctan(7x) \), so the product rule states:
\[
\frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)
\]

#### Step 1: Find the derivative of \( u(x) = x^2 \)
\[
u'(x) = 2x
\]

#### Step 2: Find the derivative of \( v(x) = \arctan(7x) \)
Using the chain rule, the derivative of \( \arctan(7x) \) is:
\[
v'(x) = \frac{1}{1+(7x)^2} \cdot \frac{d}{dx}[7x] = \frac{7}{1+49x^2}
\]

#### Step 3: Apply the product rule
\[
h'(x) = u'(x)v(x) + u(x)v'(x)
\]
Substitute the values of \( u'(x) \), \( v(x) \), and \( v'(x) \):
\[
h'(x) = 2x \cdot \arctan(7x) + x^2 \cdot \frac{7}{1+49x^2}
\]

Thus, the derivative is:
\[
h'(x) = 2x \arctan(7x) + \frac{7x^2}{1+49x^2}
\]

This is the derivative of \( h(x) = x^2 \arctan(7x) \).

Would you like further details on any of the steps?

Here are some related questions for further practice:
1. What is the derivative of \( h(x) = x^3 \arctan(2x) \)?
2. How do you apply the product rule to \( h(x) = x \sin(x) \)?
3. What is the chain rule, and when do we use it?
4. How do you find the derivative of \( f(x) = \arctan(x) \)?
5. Can you find the second derivative of \( h(x) = x^2 \arctan(7x) \)?

**Tip**: When using the product rule, always remember to differentiate both functions involved in the product separately before applying the rule!

Find the derivative of the function h(x) = x^2 arctan(7x).

Find the derivative of h(x) = x^2 arctan(7x) with a detailed solution using the product rule and chain rule. This problem covers essential calculus concepts such as trigonometric function derivatives. Ideal for students in Grades 11-12 or early university-level calculus.

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