Derivative Calculator
Calculate the numerical derivative of any function at a specific point using the central difference formula.
Derivative Calculator
Calculator
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Formula: f'(x) ≈ [f(x+h) - f(x-h)] / (2h) | f''(x) ≈ [f(x+h) - 2f(x) + f(x-h)] / h²
Worked example — f'(2) ≈ 12.00000000
Formula
f'(x) ≈ [f(x+h) - f(x-h)] / (2h) | f''(x) ≈ [f(x+h) - 2f(x) + f(x-h)] / h²
Central difference formula for numerical differentiation. The first derivative uses the symmetric difference quotient (error O(h²)). The second derivative uses the second-order central difference.
Worked Examples
Example 1: d/dx(x³) at x=2
Problem:f(x) = x³, find derivative at x = 2
Solution:f'(x) = 3x² → f'(2) = 12
Result:f'(2) ≈ 12.00000000
Example 2: Derivative of sin(x) at x=0
Problem:f(x) = sin(x), find derivative at x = 0
Solution:f'(x) = cos(x) → f'(0) = cos(0) = 1
Result:f'(0) ≈ 1.00000000
Example 3: Reciprocal function: both derivatives at x = 2
Problem:f(x) = 1/x. Find the first and second derivative at x = 2.
Solution:Analytically f'(x) = -1/x^2 and f''(x) = 2/x^3. f'(2) = -1/4 = -0.25 f''(2) = 2/8 = 0.25 Central-difference check with h = 0.0001: [f(2.0001) - f(1.9999)] / 0.0002 = (0.49997500 - 0.50002500)/0.0002 = -0.25000000 The negative first derivative says the curve is falling at x = 2; the positive second derivative says it is concave up there.
Result:f'(2) = -0.25 | f''(2) = 0.25 (falling and concave up)
Frequently Asked Questions
How do I enter functions?
Use JavaScript syntax: x*x for x², Math.sin(x), Math.exp(x), Math.log(x) for ln(x), Math.pow(x,3) for x³, Math.sqrt(x). Use * explicitly for multiplication.
What step size should I use?
Smaller h gives more accurate results, but too small causes floating-point errors. h = 0.0001 is usually a good balance. The 5-point formula is more accurate for the same h.
What is the geometric meaning of a derivative?
The derivative f'(x) at a point represents the slope of the tangent line to the curve f(x) at that specific point. If f'(x) is positive, the function is increasing at that point; if negative, it is decreasing. A zero derivative at a point indicates a local maximum, minimum, or inflection point. The second derivative f''(x) measures the rate of change of the slope — positive second derivative means the curve is concave up (opening upward like a bowl), and negative means concave down. The inflection point where the concavity changes occurs where f''(x) = 0.
What are common real-world applications of derivatives?
Derivatives are used across many fields: in physics, velocity is the derivative of position with respect to time, and acceleration is the derivative of velocity; in economics, marginal cost and marginal revenue are derivatives of total cost and total revenue functions; in engineering, derivatives help analyze stress, strain, and rate of heat transfer; in machine learning, gradient descent uses derivatives (gradients) to minimize loss functions during model training; in medicine, the rate of drug concentration change in blood is modeled using derivatives; and in finance, options pricing and risk measures like delta and gamma are all based on partial derivatives.
Why does Derivative Calculator use numerical differentiation instead of solving for an exact symbolic formula?
Symbolic differentiation requires parsing a function into an expression tree and applying calculus rules (power rule, chain rule, product rule) term by term — powerful, but complex to implement robustly for arbitrary user input. Numerical (finite-difference) differentiation instead evaluates the function at a few nearby points and estimates the slope directly, which works for virtually any computable function, including ones without a simple closed-form derivative, at the cost of a small, controllable approximation error.
What is the difference between the central difference formula and the simpler forward difference?
The forward difference approximates f'(x) as [f(x+h) − f(x)]/h, using only one side of the point — its error shrinks proportionally to h (first-order accurate). The central difference used by Derivative Calculator, [f(x+h) − f(x-h)]/(2h), averages information from both sides of the point, making its error shrink proportionally to h² (second-order accurate) — meaningfully more precise for the same step size.
How can I verify Derivative Calculator's numerical result against an exact symbolic derivative?
For common functions, compute the exact derivative by hand using standard rules (power rule: d/dx[xⁿ] = nxⁿ⁻¹; chain rule for compositions; product and quotient rules) and evaluate it at the same point. The numerical result from Derivative Calculator should match the exact value to within a tiny fraction of a percent when using the default step size — any large discrepancy usually indicates a typo in the entered function rather than a flaw in the method.
What does it mean if the derivative equals zero at a particular point?
A zero first derivative identifies a critical point — a candidate for a local maximum, local minimum, or a saddle/inflection point (like x³ at x=0). To determine which, check the second derivative at that point: positive means a local minimum, negative means a local maximum, and zero requires further (higher-order) analysis or checking values on either side.
What is the chain rule and when does a function need it?
The chain rule handles composite functions — one function nested inside another. If y = f(g(x)) then dy/dx = f'(g(x)) × g'(x): differentiate the outer function leaving the inner one intact, then multiply by the inner function's derivative. Differentiating sin(x²) gives 2x·cos(x²), not simply cos(x²). Because Derivative Calculator differentiates numerically from function values alone, it applies the chain rule implicitly and never asks you to identify the nesting.
When does a function fail to have a derivative at a point?
Differentiability requires the limit of the difference quotient to exist and be the same from both sides, so it fails at corners, cusps, vertical tangents and any discontinuity. The absolute value |x| has a corner at x = 0: the left slope is -1 and the right slope is +1. Beware that the central difference is symmetric, so at such a corner it averages the two sides and confidently reports 0. Whenever your function has a kink, jump or an unbounded slope near the point, treat the numeric output as meaningless rather than as a derivative.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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