Calculus Calculator
Interactive calculus calculator for derivatives, definite integrals, and limits. Enter a single-variable function f(x) and receive numeric estimates with transparent methods.
Use x as the variable. Supported functions include sin, cos, tan, exp, ln, sqrt, abs, powers via ^, and constants pi or e.
Derivative at x₀
Central difference
f′(x₀)
Leave empty for scaling based on x₀; set to a small non-zero value to override.
Definite integral
Composite Simpson’s rule
∫ f(x) dx
Example: π ≈ 3.14159
Higher values add precision at the cost of time.
Limit of f(x)
Samples from left and right
The calculator shrinks h automatically from this seed.
Enter inputs and click Calculate to see f′(x₀).
∫ f(x) dx will appear here.
Composite Simpson’s rule using the selected interval and subinterval count.
Left-hand limit (x → x₀⁻) will appear here.
Right-hand limit (x → x₀⁺) will appear here.
Two-sided summary will appear here.
Results sample values with decreasing h starting from 0.001.
How to use this calculator
Enter a single-variable real function f(x). Use the derivative controls to probe the slope at a point, the integral controls to compute area between limits, and the limit controls to inspect left/right behavior. Click Calculate once you have provided numeric inputs.
What this tool is for
The calculator offers transparent numeric estimates rather than symbolic proofs. It is ideal for learning, sanity-checking analytic work, and quick explorations, but always verify results in exams or graded assignments with your instructor’s allowed methods.
Methodology
Calculus hinges on differentiation (rates of change) and integration (accumulated quantities). This tool implements both numerically so you can see how computers approximate core operations.
Derivative: central difference formula
The derivative at a point is defined as the limit
Numerically we use a small finite h for the central difference.
The calculator automatically chooses h based on the scale of x₀ unless you override it, and highlights the step size in the results.
Definite integral: Simpson’s rule
Integrals accumulate area between two limits.
We partition [a, b] into n even subintervals of width \(\Delta x = \frac{b - a}{n}\) and evaluate at the grid points.
More subintervals usually mean better accuracy, but the method can still struggle with discontinuities or unbounded features.
Limits: sampling from left and right
The calculator evaluates f(x) at points approaching x₀ from smaller and larger values and averages the stable readings.
If both sides settle to nearly the same value, we report an approximate two-sided limit. If they diverge or disagree significantly, you will see a warning in the summary.
Examples you can try
- Derivative: \(f(x) = x^3 - 4x + 1\) at \(x_0 = 2\). Analytically \(f'(x) = 3x^2 - 4\), so \(f'(2) = 8\).
- Integral: \(f(x) = \sin(x)\) from 0 to \(\pi\) returns 2.
- Limit: \(f(x) = \frac{\sin(x)}{x}\) as \(x \to 0\) equals 1.
FAQ: using the calculus calculator safely
What can I do with this calculus calculator?
You can compute numeric derivatives, definite integrals, and limits with transparent methods for learning and quick checks.
How accurate are the derivative and integral results?
For smooth functions, central differences and Simpson’s rule are accurate; accuracy drops near discontinuities, highly oscillatory functions, or very wide intervals. Adjust h or n and compare with analytic work when precision matters.
Can I use this tool on exams or graded homework?
No. This is for learning and exploration. Always comply with your instructor’s rules and show the analytic work requested.
What function syntax does the calculator support?
Use x as the variable and operators +, -, *, /, ^. Functions: sin(x), cos(x), tan(x), exp(x), ln(x), sqrt(x), abs(x). Constants: pi, e.
Examples: x^2 * exp(-x), sin(x)/x, ln(1 + x), sqrt(1 - x^2) (on \([-1,1]\)).
Author
Ugo Candido creates transparent calculator logic and educational resources for real-world decision-making. Feedback and corrections are welcome.
Contact: info@calcdomain.com
Editorial policy
CalcDomain tools are designed for educational use, reviewed for clarity and accuracy, and built without paid influence on results. Inputs and assumptions are transparent so you can verify every outcome.