The formula this page uses
f′(x) = lim[h→0] (f(x + h) − f(x)) / h ∫ₐᵇ f(x) dx = F(b) − F(a)
Calculus Calculator
Explore derivatives, definite integrals, limits, roots, Taylor approximations, arc length, and average value with numerical working and a live graph.
Derivative at x = 1—Enter a function and calculate.
Show your work Step-by-step method
- The method and numerical checks will appear here.
Calculus
Calculus Calculator explained
The short version
- A derivative is a slope: how fast the output changes when the input nudges forward.
- An integral is an accumulation: the signed area trapped between a curve and the x-axis.
- They undo each other, which is why one table of rules works in both directions.
What each part means
SymbolWhat it means
f(x) — The functionThe rule being measured, for example a height in metres or a cost in dollars.
f′(x) — The derivativeThe slope of f at each x. Its units are f's units divided by x's units.
F(x) — An antiderivativeAny function whose derivative is f. Every antiderivative differs only by a constant.
a and b — The limits of integrationThe start and end of the interval you are accumulating over, in x units.
Show your work: a full example
- Start with the functionf(x) = x³ − 4x
- Power rule on x³3x²
- Power rule on −4x−4, because x¹ drops to x⁰ = 1
- Write the derivativef′(x) = 3x² − 4
- Substitute x = 2f′(2) = 3(2)² − 4 = 3(4) − 4
- Finish the arithmetic12 − 4 = 8
- Say what it meansat the point (2, 0) the tangent line rises 8 units for every 1 unit right
A second, different case
- A different case: a signed area∫₀² (x³ − 4x) dx
- Antiderivative of x³x⁴/4
- Antiderivative of −4x−2x²
- Evaluate at the top limit2⁴/4 − 2(2)² = 16/4 − 8 = 4 − 8 = −4
- Evaluate at the bottom limit0⁴/4 − 2(0)² = 0
- Subtract bottom from top−4 − 0 = −4
- Read the signx³ − 4x is below the axis for every x between 0 and 2, so the signed area is negative
Copy-ready example
f(x) = x³ − 4x
Start with the function
Questions about the Calculus Calculator
A definite integral came out negative. Did I make a mistake?
Probably not. A definite integral adds signed area, so any stretch where the curve dips below the x-axis counts as negative. If the question asks for total area instead, split the interval at each x-intercept and add the absolute values.
Why does every indefinite integral need + C?
Because differentiating kills constants. x², x² + 7 and x² − 100 all have derivative 2x, so reversing the process cannot tell which one you started from. The C carries that missing information.
What is the difference between f(2) and f′(2)?
f(2) is the height of the curve at x = 2; here it is 2³ − 4(2) = 0. f′(2) is the steepness there, which is 8. A curve can sit at height zero while climbing steeply, and this function does exactly that.
When do I need the chain rule instead of the power rule?
Whenever the thing being raised to a power is not just x. (x³ − 4x)⁵ needs the chain rule: differentiate the outside to 5(x³ − 4x)⁴ and then multiply by the inside derivative 3x² − 4.
A calculator should support the reasoning.
Use numerical results to explore and verify. For a proof or exact symbolic derivation, follow the linked step-by-step solver and check its assumptions.
Derivatives
Estimate local rate of change with centered finite differences and visualize the tangent line.
Integrals
Use adaptive Simpson integration to estimate signed accumulated area over a finite interval.
Limits
Compare values approaching from both sides so one-sided disagreement remains visible.
Taylor models
Approximate a function near a center and compare the polynomial with the original curve.
Numerical-method note: Decimal results are estimates. Discontinuities, oscillation, very large values, and poorly scaled inputs can require symbolic analysis or tighter assumptions.
Learn the mathematics
Understand calculus behind this calculator
A calculator confirms an answer. Working the method yourself is what makes the next problem faster.