Definition of the Derivative
The derivative of a function f(x) at a point x is defined as the limit as h approaches 0 of [f(x + h) - f(x)] / h. This limit represents the instantaneous rate of change of the function, or the slope of the tangent line to the curve at that point. To find the derivative, denoted f'(x) or dy/dx, start by applying this limit definition directly to the function.
Key Differentiation Rules
While the limit definition is foundational, practical computation uses rules like the power rule (d/dx [x^n] = n x^{n-1}), constant multiple rule (d/dx [c f(x)] = c f'(x)), sum rule (d/dx [f(x) + g(x)] = f'(x) + g'(x)), product rule (d/dx [f(x) g(x)] = f'(x) g(x) + f(x) g'(x)), and quotient rule (d/dx [f(x)/g(x)] = [f'(x) g(x) - f(x) g'(x)] / [g(x)]^2). These simplify finding derivatives for polynomials, products, and quotients.
Practical Example
Consider finding the derivative of f(x) = 3x^2 + 2x - 1. Apply the sum rule and power rule: the derivative of 3x^2 is 3 * 2x = 6x, of 2x is 2, and of -1 is 0. Thus, f'(x) = 6x + 2. This shows how rules break down composite functions into manageable parts.
Applications of Derivatives
Derivatives are essential in analyzing rates of change, such as velocity in physics (derivative of position) or marginal cost in economics. They enable optimization problems, curve sketching, and solving differential equations, forming the basis for advanced calculus and real-world modeling in science and engineering.