Understanding your derivative result
The sign of the second derivative, when a result is available, indicates the local concavity of the function at x0 — a standard result from calculus.
| f″(x₀) | Concavity at x₀ |
|---|---|
| Positive | Concave up — the function curves upward, like a valley, near x₀ |
| Negative | Concave down — the function curves downward, like a hill, near x₀ |
| At or near zero | Possible inflection point — concavity may be changing near x₀; further checking is needed |
- This is a numerical estimate, not a symbolic derivative — the calculator never produces a general formula f'(x); it only returns the value of the derivative at the single point x0 entered.
- Supported expression syntax: functions sin, cos, tan, asin, acos, atan, sqrt, cbrt, abs, ln (natural log), log (base-10 log), log2 (base-2 log), exp, floor, ceil, round, min, max; constants pi, e, tau; operators + − * / % ^; and the variable x. Trigonometric functions use radians.
- Functions with discontinuities, sharp corners, or undefined values near x0 (for example, evaluating near a vertical asymptote) can make the finite-difference estimate inaccurate or return no result.
- The numerical estimate has a small, generally negligible truncation error at the displayed precision for smooth, well-behaved functions, but that error can grow for functions with rapid local changes.
What is a numerical derivative?
A derivative measures the instantaneous rate of change of a function at a point — geometrically, the slope of the line tangent to the function's graph there. Symbolic differentiation produces a general formula, f'(x), valid for every x. This calculator instead computes a numerical derivative: it estimates f'(x0), a single number, at exactly the point x0 you enter, without ever producing the general formula f'(x).
The numerical method used is a 5-point central difference: the calculator evaluates f at several points very close to x0 on both sides and combines those values into a highly accurate estimate of the slope at x0, with error that shrinks proportionally to the fourth power of the step size. This is a standard, well-established technique in numerical analysis, distinct from symbolic computer-algebra differentiation.
Because the result is numerical, evaluating the derivative at a different point requires re-running the calculator with that new x0 — there is no algebraic expression for f'(x) returned that could be evaluated at other points directly.
How to use this derivative calculator
- Enter a function of x, for example x^3 - 2*x + 1, sin(x)/x, or sqrt(x) + ln(x). Multiplication must be written explicitly with *, and powers use ^.
- Enter the point x0 at which to evaluate the derivative.
- Read f'(x0), the numerical derivative at that point, and f(x0), the function's value there.
- Where the function is twice differentiable near x0, also read f''(x0), the numerical second derivative, and the tangent-line equation, which gives the best linear approximation of f near x0.
The formula behind the numerical derivative
The 5-point central difference estimate is f'(x0) ≈ [−f(x0+2h) + 8f(x0+h) − 8f(x0−h) + f(x0−2h)] / (12h), a classical finite-difference formula whose error is proportional to h⁴ — meaning halving the step size h reduces the error by roughly a factor of 16.
The second derivative is estimated with a simpler central difference: f''(x0) ≈ [f(x0+h) − 2f(x0) + f(x0−h)] / h². Its sign indicates local concavity: positive means the function curves upward (concave up) near x0, negative means it curves downward (concave down).
The tangent line at x0, the best straight-line approximation of f near that point, is y = f'(x0)(x − x0) + f(x0).
Worked example: f(x) = x³ − 2x + 1 at x0 = 2. The numerical derivative is f'(2) = 10, matching the symbolic derivative 3x² − 2 evaluated at x = 2 (3×4 − 2 = 10). The function value is f(2) = 8 − 4 + 1 = 5, so the tangent line is y = 10(x − 2) + 5.
Common mistakes
- Expecting a symbolic formula such as "3x² − 2" — this calculator returns only the numeric value of the derivative at the single point x0 you enter, using a finite-difference approximation, not symbolic (algebraic) differentiation.
- Writing implicit multiplication like 2x instead of 2*x, or using ** instead of ^ for powers — the parser requires explicit operators from the supported set.
- Evaluating at a point where the function is undefined or discontinuous, such as 1/x at x0 = 0, which makes the finite-difference estimate unreliable or returns no result.
- Assuming the numerical result is exact to arbitrary precision — the central-difference method has a small but nonzero truncation error, generally invisible at the displayed decimal places for smooth functions.
- Using log to mean natural logarithm — in this calculator's syntax, ln is the natural logarithm (base e) and log is base-10; log2 is base-2.
Perguntas frequentes
Does this calculator give me the derivative formula, like f'(x) = 3x² − 2?
No. This calculator returns only the numeric value of the derivative at the specific point x0 you enter, computed with a numerical finite-difference method. To get the derivative's behavior at another point, enter that point as the new x0 and recalculate.
What numerical method does this derivative calculator use?
A 5-point central difference: f'(x0) ≈ [−f(x0+2h) + 8f(x0+h) − 8f(x0−h) + f(x0−2h)] / (12h), a standard finite-difference formula from numerical analysis with error proportional to h⁴.
What functions and operators are supported in the expression?
sin, cos, tan, asin, acos, atan, sqrt, cbrt, abs, ln, log, log2, exp, floor, ceil, round, min, max; the constants pi, e and tau; the operators + − * / % ^; and the variable x. Trigonometric functions are evaluated in radians.
How accurate is the numerical derivative?
For smooth, well-behaved functions, the 5-point central-difference method is accurate well beyond the decimal places displayed. Accuracy degrades near discontinuities, sharp corners, or points where the function is undefined close to x0.
What does the tangent-line result mean?
The tangent line, y = f'(x0)(x − x0) + f(x0), is the best straight-line approximation of the function near x0. It touches the curve at (x0, f(x0)) and has the same slope as the function at that exact point.
Why is the second derivative sometimes missing from the results?
The second derivative uses its own central-difference formula and is only shown when that calculation returns a finite value — for example, when the function is undefined or discontinuous immediately around x0, the second-derivative estimate is omitted.
Referências
- Burden RL, Faires JD. Numerical Analysis. Cengage Learning (finite-difference approximations to derivatives).
- Press WH, Teukolsky SA, Vetterling WT, Flannery BP. Numerical Recipes: The Art of Scientific Computing. Cambridge University Press (numerical differentiation).
- Stewart J. Calculus. Cengage Learning (the derivative as instantaneous rate of change and slope of the tangent line).
- NIST Digital Library of Mathematical Functions (DLMF), Chapter 3: Numerical Methods — finite-difference formulas. dlmf.nist.gov.