How to Use the Chain Rule

Step-by-step solution

1. Name the inner function.

   $$u = 2x + 1$$

   Why: The inner function is what the power wraps around.

2. Differentiate the outer power.

   $$\frac{d}{du} u^3 = 3u^2$$

   Why: The power rule treats u as the variable for now.

3. Multiply by the derivative of the inner function.

   $$3u^2 \times 2 = 6u^2$$

   Why: The chain rule multiplies the outer derivative by the inner derivative.

4. Substitute u back.

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

   Why: Rewrite the answer in terms of x.

Practice problems

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

1. 1

   $f(x) = (x + 2)^3$

   Show answer

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

2. 2

   $f(x) = (3x - 1)^2$

   Show answer

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

3. 3

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

   Show answer

   $f^{\prime}(x) = 4x(x^2 + 1)$

4. 4

   $f(x) = (4x + 3)^3$

   Show answer

   $f^{\prime}(x) = 12(4x + 3)^2$

5. 5

   $f(x) = (x - 5)^4$

   Show answer

   $f^{\prime}(x) = 4(x - 5)^3$