Derivative Solver

A derivative represents the rate of change of a function with respect to its variable. It measures how a function's output changes as its input changes. Formal Definition (Limit Definition): f'(x) = lim(h→0) [f(x + h) - f(x)] / h Notations: • f'(x) — Lagrange notation • dy/dx — Leibniz notation • df/dx — Alternative form • Dₓf — Operator notation Geometric Interpretation: • Slope of the tangent line to the curve at a point • Instantaneous rate of change Basic Differentiation Rules: 1. Power Rule: d/dx(xⁿ) = nxⁿ⁻¹ 2. Constant Rule: d/dx(c) = 0 3. Sum Rule: d/dx[f(x) + g(x)] = f'(x) + g'(x) 4. Product Rule: d/dx[f(x)·g(x)] = f'(x)·g(x) + f(x)·g'(x) 5. Quotient Rule: d/dx[f(x)/g(x)] = [f'(x)·g(x) - f(x)·g'(x)] / [g(x)]² 6. 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 Applications: Finding maximum/minimum values, optimization, velocity and acceleration, marginal analysis in economics, tangent lines, and rate of change problems.