Second Derivative Calculator
Compute the second derivative of standard function forms to analyse concavity.
Inputs
Enter function using x as variable.
f''(x)
5.99999717
f'(x)
-1.00000000
Concavity
Concave up
Step by step
Values used
f(x) = x^4 - 3*x^2 + x; Evaluate at x = 1
Second derivative (central difference)
f''(x) ≈ [f(x+h) - 2f(x) + f(x-h)] / h²
f''(x)
= 5.99999717
f'(x)
= -1.00000000
Concavity
= Concave up
How it works
Computes the second derivative f''(x) using the finite difference formula: f''(x) ≈ [f(x+h) − 2f(x) + f(x−h)] / h². The sign of f''(x) determines concavity: positive means concave up, negative means concave down, and zero may indicate an inflection point.
Formula
Second derivative (central difference)
f''(x) ≈ [f(x+h) - 2f(x) + f(x-h)] / h²
- f(x)
- The input function
- x
- Point of evaluation
- h
- Step size (≈1e-5)
Frequently Asked Questions
What does the second derivative tell you?
It measures the rate of change of the slope. Positive means the graph curves upward (concave up), negative means it curves downward (concave down).
How do I find inflection points?
Inflection points occur where f''(x) changes sign. Evaluate the second derivative at several points to find where it crosses zero.