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The given function is \( y = 5x \sin x \), and we are asked to find the second derivative \( y'' \).

Let's start by finding the first derivative \( y' \) using the product rule. The product rule states that if \( y = uv \), then

\[
y' = u'v + uv'
\]

Here, \( u = 5x \) and \( v = \sin x \).

1. Differentiate \( u = 5x \):
   \[
   u' = 5
   \]

2. Differentiate \( v = \sin x \):
   \[
   v' = \cos x
   \]

Now, apply the product rule:

\[
y' = (5)(\sin x) + (5x)(\cos x)
\]
\[
y' = 5\sin x + 5x\cos x
\]

Next, we differentiate \( y' \) to find \( y'' \). This will again involve applying the product rule to the second term \( 5x \cos x \).

\[
y'' = \frac{d}{dx}(5\sin x) + \frac{d}{dx}(5x\cos x)
\]

For the first term \( 5\sin x \):
\[
\frac{d}{dx}(5\sin x) = 5\cos x
\]

For the second term \( 5x \cos x \), we use the product rule again, with \( u = 5x \) and \( v = \cos x \):
\[
\frac{d}{dx}(5x \cos x) = (5)(\cos x) + (5x)(-\sin x)
\]
\[
= 5\cos x - 5x\sin x
\]

Now, sum up both terms:

\[
y'' = 5\cos x + (5\cos x - 5x\sin x)
\]
\[
y'' = 10\cos x - 5x\sin x
\]

Thus, the second derivative is:

\[
y'' = 10\cos x - 5x\sin x
\]

Would you like any further details or clarifications? Here are five related questions for deeper understanding:

1. How does the product rule apply to more complex functions?
2. What happens if we have to differentiate a function like \( y = x^2 \sin x \)?
3. Can we apply the chain rule alongside the product rule in certain cases?
4. How would the differentiation process change if a constant multiplier were involved, like \( y = 3x \sin x \)?
5. What role does trigonometric identities play in simplifying derivatives of trigonometric functions?

**Tip**: When applying the product rule, make sure to carefully differentiate both parts of the product, and double-check the order of terms.

In this solution, we use the product rule to find the second derivative of the function y = 5x sin(x). The solution covers key concepts in calculus, including the product rule and the derivatives of trigonometric functions. Suitable for students in Grades 11-12.

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