Calculus Calculator
Compute numeric derivatives, definite integrals and limits of a real-valued function \(f(x)\) in one place.
For students, engineers, and self-learners
Uses standard numerical methods (central differences, Simpson’s rule) to approximate core calculus operations. Ideal for checking analytic work and building intuition.
Author: CalcDomain Math Team
Reviewed by: PhD in Applied Mathematics
Last updated: 2025
Educational use only. Always verify results analytically when precision is critical (exams, graded work, safety-critical engineering, etc.).
Interactive calculus workspace
Use x as the variable. Supported functions: sin, cos, tan, exp, ln, sqrt, abs, constants pi, e. Use ^ for powers (e.g. x^2).
Derivative at a point
Leave empty for automatic choice based on \(x_0\).
Note: this tool uses floating-point arithmetic. Very large or very small values, sharp discontinuities, or highly oscillatory functions may produce unstable results.
How this calculus calculator works
Calculus is built around two core operations: differentiation (how fast things change) and integration (how quantities accumulate). This calculator implements both numerically, which is how computers and many engineering tools work behind the scenes.
Derivative: central difference formula
For a differentiable function \( f(x) \), the derivative at a point \( x_0 \) is defined as
Numerically, we approximate this by choosing a small but finite step size \( h \):
The calculator chooses a default \( h \) based on the scale of \( x_0 \) (unless you override it). Central differences are more accurate than forward or backward differences for smooth functions because the first-order error terms cancel.
Definite integral: Simpson’s rule
The definite integral
is approximated using composite Simpson’s rule. We split \([a,b]\) into \( n \) even subintervals of width \(\Delta x = \dfrac{b-a}{n}\) and evaluate \(f(x)\) at grid points \(x_0, x_1, \dots, x_n\).
Increasing the number of subintervals \( n \) usually improves accuracy, especially when \( f(x) \) changes rapidly. For functions with discontinuities or infinite spikes, any numeric method may struggle or fail.
Limits: sampling from left and right
The limit
exists if and only if both the left-hand and right-hand limits exist and agree. The calculator evaluates \( f(x) \) at a sequence of points approaching \( x_0 \) from the left and from the right (scaled by powers of 10), then checks whether the values stabilize.
This behaves well for continuous functions and for many removable discontinuities, but it may misinterpret strongly oscillatory functions or those with very sharp features near \( x_0 \).
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 \( \pi \). The exact answer is \( 2 \).
- Limit: \( f(x) = \dfrac{\sin(x)}{x} \) as \( x \to 0 \). The limit is \( 1 \).
For more specialized topics such as Taylor and Maclaurin series, numerical differentiation schemes, or differential equations, you can use the dedicated tools in the same Core Math & Algebra family (for example, the Taylor series, Maclaurin series and differential equation calculators).
FAQ: using the calculus calculator safely
What can I do with this calculus calculator?
You can enter a single-variable real function \( f(x) \) and ask the tool to approximate: derivatives at a specified point, definite integrals over an interval, and limits (two-sided or one-sided). The goal is to give you a quick, transparent numeric check against your analytic work.
How accurate are the derivative and integral results?
For smooth, well-behaved functions, central differences and Simpson’s rule are highly accurate and widely used in applied mathematics, physics and engineering. Accuracy degrades for:
- functions with discontinuities or corners at the evaluation point,
- very large integration intervals, or
- functions with rapid oscillations (e.g. \(\sin(1/x)\) near \(x = 0\)).
If you suspect instability, try changing the step size \(h\) or the number of subintervals \(n\), and compare with analytic results when available.
Can I use this tool on exams or graded homework?
No. This calculator is provided strictly for learning support and quick checks. Always follow your instructor’s rules and show full analytic work where required. You are responsible for the correctness and appropriateness of any result you submit.
What function syntax does the calculator support?
Use x as the independent variable. Supported operations:
- Basic arithmetic: +, -, *, /, ^ for powers.
- Functions: sin(x), cos(x), tan(x), exp(x), ln(x), sqrt(x), abs(x).
- Constants: pi, e.
Examples of valid inputs include x^2 * exp(-x), sin(x)/x, ln(1 + x), sqrt(1 - x^2) (on \([-1,1]\)), and many more.
Formula (LaTeX) + variables + units
f'(x_0) = \lim_{h \to 0} \frac{f(x_0 + h) - f(x_0 - h)}{2h}.
f'(x_0) \approx \frac{f(x_0 + h) - f(x_0 - h)}{2h}.
\int_a^b f(x)\,dx
\int_a^b f(x)\,dx \approx \frac{\Delta x}{3} \left[f(x_0) + f(x_n) + 4 \sum_{k=1,3,\dots}^{n-1} f(x_k) + 2 \sum_{k=2,4,\dots}^{n-2} f(x_k) \right].
\lim_{x \to x_0} f(x)
','\
\[ f'(x_0) = \lim_{h \to 0} \frac{f(x_0 + h) - f(x_0 - h)}{2h}. \]
\[ \int_a^b f(x)\,dx \]
\[ \int_a^b f(x)\,dx \approx \frac{\Delta x}{3} \left[f(x_0) + f(x_n) + 4 \sum_{k=1,3,\dots}^{n-1} f(x_k) + 2 \sum_{k=2,4,\dots}^{n-2} f(x_k) \right]. \]
- T = property tax (annual or monthly depending on input) (currency)
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Last code update: 2026-01-19
- Initial audit spec draft generated from HTML extraction (review required).
- Verify formulas match the calculator engine and convert any text-only formulas to LaTeX.
- Confirm sources are authoritative and relevant to the calculator methodology.