Derivative Calculator

How to Use This Calculator

1. Enter your function f(x) using standard notation (e.g. x^2 + 3*x).
2. Enter the point x where you want the derivative.
3. The central difference approximation f'(x) ≈ (f(x+h)−f(x−h))/(2h) is computed instantly.
4. Use the Power Rule tab for exact polynomial derivatives.
5. Use the Professional tab for 4-method comparison and higher-order derivatives.

Formula

Example

Frequently Asked Questions

• What is a derivative? ▼
  The derivative f'(x) of a function at a point x measures the instantaneous rate of change — how fast the function value is changing at exactly that point. Geometrically, it is the slope of the tangent line to the curve y = f(x) at x. For example, if f(x) = x², then f'(x) = 2x, and at x = 3 the slope of the tangent is 6. Derivatives are used to find velocity (derivative of position), marginal cost (derivative of total cost), and optimization (maximum/minimum occur where f'(x) = 0).
• How does numerical differentiation work? ▼
  This calculator uses the central difference formula: f'(x) ≈ (f(x+h) − f(x−h)) / (2h), where h is a small step size (default h = 0.0001). By evaluating the function at two points slightly to the left and right of x, it approximates the tangent slope. This method is accurate to about 8 significant figures for smooth functions — much better than the one-sided forward difference f(x+h)−f(x))/h, which has error proportional to h rather than h². Large h gives lower precision; too small h introduces floating-point cancellation errors.
• What is the power rule? ▼
  The power rule is the most fundamental differentiation rule: for f(x) = a·xⁿ, the derivative is f'(x) = a·n·xⁿ⁻¹. Examples: d/dx[3x²] = 6x, d/dx[5x³] = 15x², d/dx[x] = 1, d/dx[7] = 0 (constant). The Power Rule tab lets you enter a polynomial term and get the exact symbolic derivative. Combined with the sum rule (derivative of a sum equals sum of derivatives), you can differentiate any polynomial term by term.
• Can this calculator find symbolic derivatives? ▼
  No — symbolic differentiation (producing formulas like 'd/dx[sin(x)] = cos(x)') requires a computer algebra system (CAS). This tool computes highly accurate numerical values of the derivative at a specific point. For exact symbolic results, use Wolfram Alpha (free, web-based), SymPy (Python library), or a CAS like Mathematica. However, the numerical approach is often all that is needed in practice — engineering and physics simulations routinely use numerical derivatives with excellent results.
• What is the second derivative? ▼
  The second derivative f''(x) is the derivative of the derivative — it measures the rate of change of the slope, which describes concavity. If f''(x) > 0 at a point, the function is concave up (shaped like a U). If f''(x) < 0, it is concave down (shaped like an inverted U). The second derivative test: if f'(c) = 0 and f''(c) > 0, then c is a local minimum; if f''(c) < 0, it is a local maximum. The calculator computes f''(x) using a second application of the central difference formula.

Related Calculators