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The given task asks to find the derivative of the function \( y = x^3 \cdot \sin{x} \).

This requires the use of the **product rule** for differentiation. The product rule states that if you have a function \( y = u(x) \cdot v(x) \), the derivative \( y' \) is:

\[
y' = u'(x) \cdot v(x) + u(x) \cdot v'(x)
\]

Here, we have:

- \( u(x) = x^3 \)
- \( v(x) = \sin{x} \)

### Step-by-step solution:

1. Differentiate \( u(x) = x^3 \):
   \[
   u'(x) = 3x^2
   \]

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

3. Apply the product rule:
   \[
   y' = u'(x) \cdot v(x) + u(x) \cdot v'(x)
   \]
   Substituting the values:
   \[
   y' = 3x^2 \cdot \sin{x} + x^3 \cdot \cos{x}
   \]

Thus, the derivative of the function is:
\[
y' = 3x^2 \sin{x} + x^3 \cos{x}
\]

Let me know if you'd like any further clarification or details on this process!

Here are five related questions:

1. How do you apply the product rule to more complex functions?
2. What is the derivative of \( \sin{x} \cdot \cos{x} \)?
3. How can you use the chain rule in conjunction with the product rule?
4. What is the derivative of the function \( x^4 \cdot e^x \)?
5. How does the product rule extend to three or more functions?

**Tip:** When dealing with trigonometric functions, always check if you can simplify the problem using identities before differentiating.

Знайдіть похідну функції y = x^3 · sin(x)

Learn how to find the derivative of y = x^3 · sin(x) using the product rule. This solution breaks down each step, applying key calculus concepts like the product rule and derivatives of trigonometric functions.

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