How to Use the Product Rule

Step-by-step solution

1. Label the two factors.

   $$u = x^2,\quad v = x + 3$$

   Why: The product rule needs each factor and its derivative.

2. Differentiate each factor.

   $$u^{\prime} = 2x,\quad v^{\prime} = 1$$

   Why: The power rule gives u prime, and the derivative of x + 3 is 1.

3. Write the product rule.

   $$f^{\prime}(x) = u^{\prime} v + u v^{\prime}$$

   Why: Each factor takes a turn being differentiated.

4. Substitute and simplify.

   $$f^{\prime}(x) = 2x(x + 3) + x^2(1) = 2x^2 + 6x + x^2 = 3x^2 + 6x$$

   Why: Combining like terms gives the final derivative.

Practice problems

Use the same method. Work each problem on paper, then open the answer to check.

1. 1

   $f(x) = x(x + 1)$

   Show answer

   $f^{\prime}(x) = 2x + 1$

2. 2

   $f(x) = x^2(x + 2)$

   Show answer

   $f^{\prime}(x) = 3x^2 + 4x$

3. 3

   $f(x) = x^3(x - 1)$

   Show answer

   $f^{\prime}(x) = 4x^3 - 3x^2$

4. 4

   $f(x) = x(x^2 + 2)$

   Show answer

   $f^{\prime}(x) = 3x^2 + 2$

5. 5

   $f(x) = x^2(x^2 + 1)$

   Show answer

   $f^{\prime}(x) = 4x^3 + 2x$