Differentiate a function step by step, then use the integral and limit calculators for the rest of the toolkit.
What this calculator does
Calculus rests on two operations that undo each other: differentiation, which finds the rate of change, and integration, which accumulates it. This page handles the differentiation side and links to the others.
The second and third derivatives are shown alongside the first, which is what you need for concavity, inflection points and the second-derivative test.
How to use it
- Enter a function of x.
- Read the first, second and third derivatives.
- Evaluate at a point to check turning points and concavity.
How the number is calculated
f′(x) gives slope · f″(x) gives concavity
f′(x) = 0 locates turning points · f″(x) = 0 locates inflection points
The default scenario, worked through
The calculator opens with a function f of x^3 – 5x^2 + 2x – 7. On those figures the derivative is 3·x^2-10·x+2. Underneath, f comes out at x^3-5·x^2+2·x-7 and f′ at 3·x^2-10·x+2. Change any field and every figure updates as you type.
FAQ
How do I find turning points?
Set the first derivative to zero. Use the second derivative at those points — negative means a maximum, positive a minimum.
What is the fundamental theorem of calculus?
It states that integration and differentiation are inverse operations: the definite integral of f′ from a to b equals f(b) − f(a).