f(a)
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Math & numbers · Calculus
Derivative
Derivative
calculator
Enter any function and a point, and get the slope f′(a), the value f(a) and the tangent line — computed in the page as you type, with nothing sent anywhere.
d/dxcentral difference · h = 10⁻⁶
f′(2) of x^3
12
Tangent line
y = 8 + 12(x − 2)
f′(a) ≈ [ f(a+h) − f(a−h) ] / 2h
The slope of any function, at any point
A derivative answers one question: if x nudges a little, how fast does f(x) respond? This tool gives that answer numerically — type a function, pick a point, and it reports f′(a) together with f(a) and the full tangent-line equation, recomputing on every keystroke.
How it is computed
f′(a) ≈ [ f(a+h) − f(a−h) ] / 2h, h = 10⁻⁶
central difference · error shrinks with h²
central difference · error shrinks with h²
The central difference straddles the point symmetrically, which cancels the leading error term — for smooth functions the result is typically accurate to six or more decimal places. The same function library as the rest of the site is available: sqrt, exp, ln, log, the trigonometric and hyperbolic families, abs, powers with ^, factorials with !, and the constants pi, e and tau.
Worked example
f(x) = x^3 at a = 2
f(2) = 8 · f′(2) = 12 · tangent: y = 8 + 12(x − 2)
f(2) = 8 · f′(2) = 12 · tangent: y = 8 + 12(x − 2)
One honest limit: this is numerical, not symbolic. It tells you the slope of sin(x) at 0 is 1; it does not tell you the derivative is cos(x). At corners and jumps — abs(x) at 0, for instance — a two-sided difference reports the average of the two one-sided slopes, so treat non-smooth points with care. For multi-step working with variables, the advanced calculator runs the same engine in notebook form, and the integral calculator is its mirror image: area instead of slope.
Common questions
Derivative FAQ
No — it is numerical. It evaluates the function just either side of your point and reports the slope, accurate to around six or more decimal places for smooth functions. It will not output an expression like cos(x); it outputs the number that expression equals at your chosen point.
Anything the site's engine parses: polynomials, sqrt, cbrt, exp, ln, log, log2, sin, cos, tan and their inverses, the hyperbolic family, abs, powers with ^, factorial with !, and the constants pi, e and tau. Combine them freely, for example ln(x^2 + 1) or e^(−x)*sin(x).
Because straddling the point symmetrically cancels the largest error term. A one-sided difference has error proportional to h; the central difference's error is proportional to h², which at h = 10⁻⁶ is far smaller than anything you would notice at eight displayed decimal places.
At a corner such as abs(x) at 0, the two one-sided slopes disagree and the central difference reports their average — 0 in that case — even though no true derivative exists there. At a jump the result can be huge or unstable. The tool cannot detect these situations for you, so know your function.
It is the best straight-line approximation to your function at that point: y = f(a) + f′(a)(x − a). It is what linear approximation, Newton's method and most of first-year calculus are built on, and seeing it written out makes the derivative concrete rather than abstract.
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