The Second Derivative Calculator evaluates the derivative of a function twice. The second derivative is particularly useful because it describes how the slope is changing and helps identify concavity and possible turning behavior.
Meaning of the Second Derivative
For a function f(x), the second derivative is f″(x) = d²f/dx². If f″(x) is positive over an interval, the graph is concave upward there; if it is negative, the graph is concave downward. A change in concavity can occur at an inflection point.
How to Use the Calculator
Enter the function and any evaluation point requested by the tool. First differentiate the original function conceptually, then differentiate that result once more. This makes it easier to recognize whether the calculator has interpreted the expression correctly.
Example
For f(x) = x³ − 4x, the first derivative is 3x² − 4 and the second derivative is 6x. At x = 1, the second derivative is 6, indicating upward concavity at that point.
Second Derivative and Extrema
At a critical point where f′(x) = 0, a positive second derivative supports a local minimum and a negative second derivative supports a local maximum. If the second derivative is zero, this test is inconclusive and another method is needed.
Useful Check
Do not use the sign of the second derivative alone to locate every maximum or minimum. Find the critical points first, then use the second derivative or another appropriate test.