Derivative Calculator
Solve derivative problems step-by-step with our free calculator. See formulas, worked examples, and clear explanations.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator
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
Frequently Asked Questions
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.
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.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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