Differentiation Calculator

Differentiation is the process of finding the derivative of a function, which represents the instantaneous rate of change. Core Definition: The derivative of f(x) measures how f(x) changes as x changes. Formal Definition (Limit Definition): f'(x) = lim(h→0) [f(x + h) - f(x)] / h Notations for Derivatives: • f'(x) — Lagrange notation (prime notation) • dy/dx — Leibniz notation (differential notation) • df/dx — Alternative Leibniz form • ḋ, ẍ — Newton notation (dots, rarely used today) • Dₓf — Operator notation Geometric Interpretation: The derivative at a point represents: • Slope of the tangent line to the curve at that point • Instantaneous rate of change • How fast y changes relative to x Basic Differentiation Rules: 1. Power Rule: d/dx(xⁿ) = nxⁿ⁻¹ Example: d/dx(x³) = 3x² 2. Constant Rule: d/dx(c) = 0 (c is a constant) 3. Constant Multiple Rule: d/dx(cf(x)) = c·f'(x) 4. Sum/Difference Rule: d/dx[f(x) ± g(x)] = f'(x) ± g'(x) 5. Product Rule: d/dx[f(x)·g(x)] = f'(x)·g(x) + f(x)·g'(x) 6. Quotient Rule: d/dx[f(x)/g(x)] = [f'(x)·g(x) - f(x)·g'(x)] / [g(x)]² 7. Chain Rule: d/dx[f(g(x))] = f'(g(x))·g'(x) Common Derivatives: • d/dx(sin x) = cos x • d/dx(cos x) = -sin x • d/dx(eˣ) = eˣ • d/dx(ln x) = 1/x • d/dx(aˣ) = aˣ ln a Higher-Order Derivatives: • First derivative: f'(x) or dy/dx — rate of change • Second derivative: f''(x) or d²y/dx² — rate of change of rate (acceleration) • nth derivative: f⁽ⁿ⁾(x) or dⁿy/dxⁿ Types of Differentiation: 1. Ordinary Differentiation: Functions of one variable 2. Partial Differentiation: Functions of multiple variables (∂f/∂x) 3. Implicit Differentiation: When y is not explicitly solved for 4. Logarithmic Differentiation: Using ln to simplify complex products/quotients Applications: Finding maximum and minimum values, optimization problems, velocity and acceleration in physics, marginal cost and revenue in economics, rate of reaction in chemistry, population growth rates, tangent lines, curve sketching, and related rates problems.