Average rate of change: the slope of a straight line doesn't fit a curve
The slope of a straight line is constant everywhere along it, calculated as rise over run: the change in the vertical value divided by the change in the horizontal value between any two points on the line. A curve, by contrast, has a slope that changes continuously from point to point — the graph of f(x) = x^2 is shallow near x = 0 and steep further out, so no single slope value can describe the whole curve.
The average rate of change of a function between two points a and b is calculated the same way a line's slope is: [f(b) − f(a)] / (b − a). This number describes how much the function changed, on average, over that entire interval, but it does not describe how steep the function is at any one specific point inside that interval — a curve can rise quickly at one end of an interval and slowly at the other while still having the same average rate of change throughout.
From secant line to tangent line: the limit definition of a derivative
A line drawn through two points on a curve is called a secant line, and its slope equals the average rate of change between those two points, [f(x+h) − f(x)] / h, where h is the horizontal distance between them. As the second point is moved closer and closer to the first — as h shrinks toward zero — the secant line rotates and approaches a limiting position called the tangent line, the line that just touches the curve at a single point and matches its direction there.
The derivative of f at x is defined as the slope of this tangent line, expressed formally as the limit of the secant slope as h approaches zero: f'(x) = lim(h→0) [f(x+h) − f(x)] / h. This is the formal definition taught in every standard calculus course, and it converts the vague idea of "instantaneous rate of change" into a precise limit that can be evaluated or approximated.
Numerical vs symbolic differentiation: two different questions
Symbolic differentiation applies algebraic rules — the power rule, product rule, chain rule, and others — directly to a function's formula to produce a new formula, f'(x), valid at every x where the original function is differentiable. Given f(x) = x^2, symbolic differentiation returns the general formula f'(x) = 2x, which can then be evaluated at any point: at x = 3 it gives 6, at x = 10 it gives 20, without any further calculus.
Numerical differentiation answers a narrower question: what is the derivative's value at one specific point, x0, without ever deriving the general formula? This site's derivative calculator uses a 5-point central difference, a standard numerical-analysis technique that 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 alone. The two approaches are not competing methods for the same output — one produces a formula, the other produces a single number at a single point — and confusing them is the most common misunderstanding when reading a derivative calculator's result.
| Approach | What it returns | Example for f(x) = x² |
|---|---|---|
| Symbolic differentiation | A general formula, f'(x), valid at every x | f'(x) = 2x |
| Numerical differentiation (5-point central difference) | A single number, the estimated value of f'(x0) at one chosen point | f'(3) ≈ 6 |
Worked example: the derivative of x² at x = 3
Symbolically, f(x) = x^2 has derivative f'(x) = 2x by the power rule, so f'(3) = 2 × 3 = 6 exactly. Numerically, the 5-point central difference formula estimates f'(x0) ≈ [−f(x0+2h) + 8f(x0+h) − 8f(x0−h) + f(x0−2h)] / (12h) for a small step size h; using h = 0.000001 and x0 = 3, this numerical formula evaluates to approximately 6.000000001, matching the exact symbolic result to eight decimal places.
This agreement is expected, not coincidental: for a smooth function like x^2, the central-difference method's error shrinks proportionally to h^4, so a very small h produces a numerical estimate that is effectively indistinguishable from the exact symbolic answer at the precision normally displayed. The two methods answer the same underlying mathematical question — the slope of the tangent line at x = 3 — by two different routes, one algebraic and exact, the other numerical and approximate but extremely accurate for well-behaved functions.
Why the distinction matters when reading a calculator's result
A numerical derivative calculator, including the one on this site, never outputs a formula like 2x — it outputs one number, the estimated slope at the single x0 entered. Finding the derivative's value at a different point requires re-entering that new point and recalculating, because no general expression for f'(x) is produced or stored along the way.
This distinction has a practical consequence: numerical differentiation can evaluate the derivative of functions that are difficult or impossible to differentiate symbolically in closed form — for example, functions built from data, simulations, or expressions without a simple algebraic derivative — because it only ever needs to evaluate f at nearby points, not to manipulate its formula. The tradeoff is that each result is a single-point estimate rather than a reusable general formula.
Frequently asked questions
What is a derivative in simple terms?
A derivative is the instantaneous rate of change of a function at a specific point, equivalent to the slope of the line tangent to the function's graph at that point. It is formally defined as the limit of the average rate of change (the slope of a secant line) as the two points defining that secant line move together into one.
What is the limit definition of a derivative?
The derivative of f at x is defined as f'(x) = lim(h→0) [f(x+h) − f(x)] / h, the limit of the secant-line slope between the points x and x+h as h shrinks toward zero. This limit converts the informal idea of instantaneous rate of change into a precise mathematical definition.
What is the difference between numerical and symbolic differentiation?
Symbolic differentiation applies algebraic rules to a function's formula to produce a new general formula, f'(x), valid at every point. Numerical differentiation instead estimates the derivative's value at one specific point using nearby function evaluations, such as a 5-point central difference, without ever producing a general formula.
Why does a numerical derivative calculator only give one number instead of a formula?
Because numerical differentiation works by evaluating the function at points very close to the one point requested and combining those values into a slope estimate — it never manipulates the function's algebraic structure, so it has no way to produce a formula valid at other points. To evaluate the derivative elsewhere, the new point must be entered and the calculation repeated.
What is the derivative of x² at x = 3?
Exactly 6. Symbolically, the power rule gives f'(x) = 2x for f(x) = x², so f'(3) = 6. Numerically, a 5-point central difference with a small step size produces the same result to many decimal places, since the two methods compute the same underlying quantity — the slope of the tangent line at x = 3 — by different routes.
References
- Stewart J. Calculus. Cengage Learning (the derivative as the limit of a difference quotient and the slope of the tangent line).
- Burden RL, Faires JD. Numerical Analysis. Cengage Learning (finite-difference approximations to derivatives).
- Spivak M. Calculus (4th ed). Publish or Perish, 2008 (rigorous treatment of limits and the derivative).
- NIST Digital Library of Mathematical Functions (DLMF), Chapter 3: Numerical Methods. dlmf.nist.gov.