Derivatives

The derivative measures how a function changes at any instant. It is the foundation of calculus and is used to find rates of change, optimize values, and locate critical points.

Basic Derivative Rules

Power rule: d/dx(xⁿ) = n·xⁿ⁻¹

Constant rule: d/dx(c) = 0

Constant multiple rule: d/dx(c·f(x)) = c·f'(x)

Sum rule: d/dx(f(x) + g(x)) = f'(x) + g'(x)

Derivative of Polynomial Functions Apply the power rule term by term.

Example: d/dx(3x⁴ - 2x² + 5) = 12x³ - 4x

Maximum Values of Quadratic Functions For a quadratic f(x) = ax² + bx + c, the vertex is at x = -b/(2a). If a < 0, the vertex is a maximum; if a > 0, it is a minimum.

Example: f(x) = −x² + 6x has its maximum at x = −6/(2·−1) = 3.

Critical Points Critical points occur where the derivative is zero or undefined. They are candidates for maxima, minima, or inflection points.

Applications Derivatives model rates such as reaction rates, velocity, and rates of change in populations.