How to Find the Second Derivative

Step-by-step solution

1. Find the first derivative.

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

   Why: The power rule differentiates each term once.

2. Differentiate again.

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

   Why: The second derivative is the derivative of the first derivative.

3. Read what it means.

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

   Why: The second derivative measures how the slope itself changes.

Practice problems

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

1. 1

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

   Show answer

   $f^{\prime\prime}(x) = 6x + 4$

2. 2

   $f(x) = x^4 - 2x^2$

   Show answer

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

3. 3

   $f(x) = 2x^3 - 6x$

   Show answer

   $f^{\prime\prime}(x) = 12x$

4. 4

   $f(x) = x^3 - 6x^2 + 5x$

   Show answer

   $f^{\prime\prime}(x) = 6x - 12$

5. 5

   $f(x) = 5x^2 - 3x$

   Show answer

   $f^{\prime\prime}(x) = 10$