Derivative & Integral Calculator
Find the value of a derivative at a point, or a definite integral over an interval. This is a numerical calculator: it returns accurate numbers, not formulas. The derivative of x² at x = 3 comes back as 6, not as 2x — which is the right tool for checking a value or finding an area, and the wrong one if you need the symbolic answer.
Differentiate or integrate
This calculator is numerical, not symbolic
It does not return a formula. Ask for the derivative of x² and you get 6 at x = 3, not 2x. Ask for an integral and you get a number for a definite integral, not an antiderivative with a constant of integration.
That is a real limitation and worth knowing before you rely on it. For homework asking you to differentiate or integrate an expression, you need the symbolic answer and this will not give it. For checking a value you have worked out by hand, finding a gradient at a point, or getting the area under a curve that has no elementary antiderivative, a numerical answer is exactly what you want — and it works on functions no symbolic tool can integrate in closed form.
How to use this calculator
- Choose a derivative at a point, or a definite integral over an interval.
- Type f(x) using x as the variable.
- Enter the point, or the two limits.
- Press Calculate. The derivative view also gives the second derivative, the tangent line and whether the curve is concave up or down there.
The methods, and how accurate they are
Derivatives use a five-point central difference, which is accurate to fourth order in the step size, rather than the two-point formula taught first. The step is scaled by the cube root of machine epsilon: too large and the formula is a poor approximation to the limit, too small and subtracting two nearly equal values destroys the precision you were trying to gain. Typical agreement with the exact answer is ten or eleven significant figures.
Integrals use adaptive Simpson's rule. The interval is split, each half estimated, and where the two estimates disagree by more than the tolerance that part is subdivided again. Smooth stretches are therefore cheap and awkward ones get the effort. The tolerance is 1×10−10.
Both are approximations. They are extremely good ones for well behaved functions, and you should treat the last digit or two as uncertain rather than exact.
What it will refuse
An integral whose function blows up inside the interval — 1/x across zero, for instance — is an improper integral and needs a limit process this calculator does not perform. Rather than return a confident number from sampling either side of the singularity, it checks the interval first and tells you where the problem is. Infinite limits are not supported for the same reason.
Frequently Asked Questions
Does this give me the derivative as a formula?
No. It is numerical, so it gives the value of the derivative at a point you choose. The derivative of x squared at x = 3 comes back as 6, not as the expression 2x. If you need the symbolic form, this is not the right tool.
How accurate are the answers?
Derivatives use a five-point central difference and typically agree with the exact value to ten or eleven significant figures. Integrals use adaptive Simpson's rule to a tolerance of 1e-10. Both are approximations, so treat the final digit or two as uncertain.
Can it calculate the area under a curve?
Yes, that is exactly what the definite integral option does. Note that it returns the signed area: a region below the x-axis counts as negative, so the integral of x cubed from -2 to 2 is zero. For the total unsigned area, integrate the absolute value with abs(x^3).
Why does it refuse to integrate 1/x from -1 to 1?
Because the function is infinite at x = 0, which is inside the interval. That is an improper integral and needs a limit process this calculator does not perform. Returning a number from sampling either side of the singularity would be misleading, so it reports the problem instead.
Should trigonometric functions be in degrees or radians?
Radians, in almost all calculus. The familiar rules, including that the derivative of sin(x) is cos(x), only hold in radians. The degrees option is there for the occasional applied problem stated that way, and it is off by default.