Derivative Calculator
Calculate derivatives of polynomial, exponential, logarithmic, and trigonometric functions.
What is a derivative and what does it represent?
A derivative measures the rate of change or slope of a function at any point. d/dx represents "change in y / change in x" instantaneously. Example: Position function -> derivative = velocity (rate of position change). Velocity function -> derivative = acceleration (rate of velocity change). f(x) = x^2 -> f'(x) = 2x means slope at x=3 is 6. Derivatives are fundamental to calculus, physics, economics, optimization.
What are the basic derivative rules?
Power rule: d/dx(xⁿ) = nxⁿ⁻^1. Constant: d/dx(c) = 0. Sum: d/dx(f+g) = f' + g'. Product: d/dx(fg) = f'g + fg'. Quotient: d/dx(f/g) = (f'g - fg')/g^2. Chain rule: d/dx(f(g(x))) = f'(g(x))·g'(x). Examples: d/dx(3x^2) = 6x, d/dx(x^3 + 2x) = 3x^2 + 2, d/dx(5) = 0.
What are derivatives of common functions?
Memorize these: d/dx(xⁿ) = nxⁿ⁻^1, d/dx(eˣ) = eˣ, d/dx(ln x) = 1/x, d/dx(sin x) = cos x, d/dx(cos x) = -sin x, d/dx(tan x) = sec^2x, d/dx(aˣ) = aˣ ln a. Examples: d/dx(e^2ˣ) = 2e^2ˣ (chain rule), d/dx(x⁵) = 5x⁴, d/dx(sqrtx) = 1/(2sqrtx). These form basis for all differentiation.
When do you use derivatives in real life?
Optimization: Maximize profit, minimize cost (set derivative = 0). Physics: Velocity, acceleration, force. Economics: Marginal cost, marginal revenue. Engineering: Rate of heat transfer, fluid flow. Medicine: Drug absorption rates. Machine learning: Gradient descent optimization. Example: Profit P(x) = -x^2 + 100x. P'(x) = -2x + 100 = 0 → x = 50 maximizes profit.