The product rule is a fundamental technique in calculus used to find the derivative of the product of two functions. For two functions \(u(x)\) and \(v(x)\), the product rule states:

\[
(uv)' = u'v + uv'
\]

This can be extended to solve differential equations involving products of functions. Here are some examples and explanations:

1. **Basic Product Rule Application**:
   If you have a product of two functions \(u(x)\) and \(v(x)\), the derivative with respect to \(x\) is:
   \[
   \frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)
   \]

2. **Example with Specific Functions**:
   Suppose \(u(x) = x^2\) and \(v(x) = \sin(x)\). Using the product rule:
   \[
   \frac{d}{dx}[x^2 \sin(x)] = \frac{d}{dx}[x^2] \cdot \sin(x) + x^2 \cdot \frac{d}{dx}[\sin(x)]
   \]
   Calculating each term:
   \[
   \frac{d}{dx}[x^2] = 2x
   \]
   \[
   \frac{d}{dx}[\sin(x)] = \cos(x)
   \]
   Therefore:
   \[
   \frac{d}{dx}[x^2 \sin(x)] = 2x \sin(x) + x^2 \cos(x)
   \]

3. **Using the Product Rule in Differential Equations**:
   Consider a differential equation where the product rule is applied. For example:
   \[
   \frac{d}{dx}[u(x)v(x)] = f(x)
   \]
   You can solve for \(u(x)v(x)\) by integrating both sides:
   \[
   u(x)v(x) = \int f(x) \, dx
   \]

4. **Second Derivative Using Product Rule**:
   If you need the second derivative of the product of two functions, you apply the product rule twice. For example, if \(y = uv\):
   \[
   \frac{d^2}{dx^2}[uv] = \frac{d}{dx}[u'v + uv'] = u''v + u'v' + u'v' + uv''
   \]
   Simplifying, we get:
   \[
   \frac{d^2}{dx^2}[uv] = u''v + 2u'v' + uv''
   \]

5. **Example in Solving a Differential Equation**:
   Consider a differential equation involving a product:
   \[
   \frac{d}{dx}[y e^x] = x^2
   \]
   Applying the product rule:
   \[
   y' e^x + y e^x = x^2
   \]
   Simplifying and solving for \(y\):
   \[
   y' e^x + y e^x = x^2
   \]
   \[
   e^x(y' + y) = x^2
   \]
   Divide both sides by \(e^x\):
   \[
   y' + y = x^2 e^{-x}
   \]
   This is now a first-order linear differential equation that can be solved using an integrating factor.

Would you like more details or have any specific questions about using the product rule in differential equations? Here are some related questions you might find useful:

1. How do you solve first-order linear differential equations?
2. What is an integrating factor and how is it used in solving differential equations?
3. Can you provide more examples of applying the product rule in calculus?
4. How does the product rule extend to higher-order derivatives?
5. What are some common applications of the product rule in physics and engineering?

**Tip:** When applying the product rule, always ensure to differentiate both functions involved in the product and remember to sum the resulting terms to obtain the correct derivative.

Explore how to apply the product rule in differential equations with detailed examples and explanations. Learn how to find derivatives of products of functions and solve differential equations involving product rules effectively. Suitable for advanced high school and college-level students.

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