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The problem asks to find \(\frac{d^2y}{dx^2}\) for the given function:

\[
y = (3x^2 + 11)\sin(8x)
\]

This involves calculating the second derivative. Let’s break it down:

1. **First Derivative**: We apply the product rule since this is the product of two functions: \((3x^2 + 11)\) and \(\sin(8x)\).

The product rule is:

\[
\frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)
\]

Where:
- \( u(x) = 3x^2 + 11 \)
- \( v(x) = \sin(8x) \)

First, we compute the derivatives of \( u(x) \) and \( v(x) \):
- \( u'(x) = \frac{d}{dx}(3x^2 + 11) = 6x \)
- \( v'(x) = \frac{d}{dx}[\sin(8x)] = 8\cos(8x) \)

Now applying the product rule:

\[
\frac{dy}{dx} = (6x)\sin(8x) + (3x^2 + 11)(8\cos(8x))
\]

Simplifying:

\[
\frac{dy}{dx} = 6x\sin(8x) + 8(3x^2 + 11)\cos(8x)
\]

2. **Second Derivative**: We need to differentiate \(\frac{dy}{dx}\) again. Again, we will apply the product rule to each term separately.

- For the first term \( 6x\sin(8x) \), we apply the product rule:
  \[
  \frac{d}{dx}[6x\sin(8x)] = 6\sin(8x) + 48x\cos(8x)
  \]

- For the second term \( 8(3x^2 + 11)\cos(8x) \), we again apply the product rule:
  \[
  \frac{d}{dx}[8(3x^2 + 11)\cos(8x)] = 8(6x)\cos(8x) - 64(3x^2 + 11)\sin(8x)
  \]

Now, combining everything:

\[
\frac{d^2y}{dx^2} = 6\sin(8x) + 48x\cos(8x) + 48x\cos(8x) - 64(3x^2 + 11)\sin(8x)
\]

Simplifying further:

\[
\frac{d^2y}{dx^2} = 6\sin(8x) + 96x\cos(8x) - 64(3x^2 + 11)\sin(8x)
\]

This is the second derivative \(\frac{d^2y}{dx^2}\).

Would you like any further details or clarifications?

Here are 5 related questions to expand your understanding:
1. How do you apply the product rule for higher-order derivatives?
2. What would change if the function involved a different trigonometric function, like \(\cos(8x)\)?
3. Can you differentiate any composite functions involving chain and product rules?
4. How would you approach the problem if asked to find the third derivative?
5. How can we simplify the expression for the second derivative even further?

**Tip**: Always make sure to differentiate both functions involved in a product separately when applying the product rule.

Find the second derivative of y = (3x^2 + 11)sin(8x).

Learn how to find the second derivative of the function y = (3x^2 + 11)sin(8x) using the product rule. This detailed solution breaks down the first and second derivatives step-by-step, involving differentiation techniques and trigonometric functions.

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