What is a derivative in calculus?

The derivative of a function at a point measures the instantaneous rate of change of the function at that point. Geometrically, it is the slope of the tangent line to the graph of the function. Formally, the derivative of f at x0 is defined as the limit of the difference quotient as the step size tends to zero: f'(x0) = lim(h→0) [f(x0 + h) − f(x0)] / h, when this limit exists.

Is this calculator symbolic or numerical?

This calculator is numerical. It approximates derivatives using finite difference formulas (primarily a central difference scheme) rather than producing an exact symbolic expression for f'(x). It is ideal for intuition, checking hand calculations and working with functions that are difficult to differentiate analytically. For step-by-step symbolic derivatives you should still use a CAS or perform the differentiation by hand.

How accurate is the numerical derivative?

For smooth functions and a reasonable choice of step size h, central difference formulas can be very accurate. The calculator computes the derivative using h and again using h/2, then uses the difference as a simple error indicator. Very small h values can amplify rounding errors, while very large h values give coarse approximations, so interpreting the error estimate and possibly adjusting h is important.

Can this calculator handle non-differentiable points?

At points where the derivative does not exist (for example sharp corners such as f(x) = |x| at x = 0 or discontinuities) finite difference formulas may still produce finite numbers, but the left-hand and right-hand slopes can differ significantly or behave erratically as h changes. The calculator helps you spot such cases by providing an error estimate and allowing you to inspect f(x) and f'(x) on an interval, but mathematical judgement is still needed.

Which syntax can I use for functions?

Use x as the variable and the operators +, -, *, / and ^ for powers, together with sin, cos, tan, asin, acos, atan, ln, log, sqrt, abs, exp, floor, ceil, max, min, and constants pi and e. Trigonometric functions use radians, consistent with typical scientific software and programming languages.

Derivative Calculator (Numerical)

f'(x) at a point & on an interval

Approximate the derivative of a real-valued function using finite differences. This tool computes f(x₀), f'(x₀) with a central difference scheme, estimates numerical error and lets you inspect f(x) and f'(x) across an interval in a detailed table.

Focus: high-quality numerical derivatives for intuition, verification and applied work. For step-by-step algebraic differentiation and symbolic expressions, use a CAS – then cross-check here.

1. Define the function and derivative point

Function f(x)

Use x as the variable. Allowed: +, -, *, /, ^, sin, cos, tan, asin, acos, atan, ln, log, sqrt, abs, exp, floor, ceil, max, min, constants pi, e. Trig functions use radians.

Point x₀

The point at which to evaluate f(x₀) and f'(x₀).

Step size h

Small positive number (e.g. 1e-4). The calculator also tests h/2 for error.

2. Interval for derivative table (optional)

Interval start a

Start of [a, b] for the sample table.

Interval end b

End of [a, b]; must be different from a.

Sample points on [a, b]

Number of (x, f(x), f'(x)) rows in the table.

Decimals

Display precision for all numerical results.

Quick examples:

3. Numerical results at x₀

f(x₀)

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Derivative f'(x₀) (h)

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Derivative f'(x₀) (h/2)

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Estimated error |h − h/2|

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Scheme used

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Stability note

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Warnings

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4. Sample table on [a, b]

Inspect how the function and its numerical derivative behave across the interval. This is useful for spotting non-differentiable points, sign changes of the slope and regions of rapid variation.

x	f(x)	Approx. f'(x)

Definition and meaning of the derivative

The derivative of a function at a point measures its instantaneous rate of change or, geometrically, the slope of the tangent line to the graph at that point. For a real function \(f(x)\), the derivative at \(x_0\) is formally defined as:

\[ f'(x_0) = \lim_{h \to 0} \frac{f(x_0 + h) - f(x_0)}{h}, \]

whenever this limit exists. If the limit exists for every point in an interval, we obtain a new function \(f'(x)\), the derivative of \(f\).

In applications, derivatives describe velocities, accelerations, growth rates, slopes of curves, marginal costs, sensitivities in models and much more.

Numerical derivatives with finite differences

Because we cannot take an actual limit on a computer, numerical derivatives approximate the definition of the derivative using a small step h. A popular and accurate choice is the central difference formula:

\[ f'(x_0) \approx \frac{f(x_0 + h) - f(x_0 - h)}{2h}. \]

This uses information on both sides of \(x_0\) and has a truncation error proportional to \(h^2\) for smooth functions. Our calculator also compares this approximation with the same formula at step size \(h/2\) to estimate numerical error.

Forward and backward differences

Near boundaries or where the function is undefined on one side of \(x_0\), a central difference may not be possible. In those cases we fall back to one-sided schemes:

Forward difference: \[ f'(x_0) \approx \frac{f(x_0 + h) - f(x_0)}{h}. \]
Backward difference: \[ f'(x_0) \approx \frac{f(x_0) - f(x_0 - h)}{h}. \]

These are less accurate than central differences (error typically proportional to \(h\)), but still useful when only one side of the point is available.

FAQ: using this derivative calculator effectively

How should I choose the step size h?

There is a trade-off: if h is too large, the approximation is coarse and misses local behaviour; if h is too small, rounding errors and cancellation can dominate. Values like 10⁻³ to 10⁻⁵ are often a good starting point for smooth functions with moderate magnitudes. The error estimate |f'ₕ − f'ₕ₍₂₎| reported by the calculator helps you judge whether h is appropriate.

What if the derivative does not exist at a point?

At non-differentiable points (sharp corners, cusps, jumps) finite difference formulas can still output numerical values, but those values can be unstable with respect to h and may differ significantly between left and right. In such cases, use the sample table and the stability note as clues, and always confirm by analysing the function definition directly.

Can I use this in exams or for high-stakes engineering decisions?

This tool is designed as a didactic and exploratory aid, and for quick checks in applied work. For exam solutions, formal proofs or safety-critical engineering, you should always complement numerical derivatives with analytic methods or validated scientific computing libraries.