Definition of Derivatives
In mathematics, a derivative measures how a function changes at a specific point, representing the slope of the tangent line to the function's graph. Formally, the derivative of a function f(x) at a point x = a is defined as the limit as h approaches 0 of [f(a + h) - f(a)] / h. This quantifies the instantaneous rate of change, distinguishing it from average rates over intervals.
Key Principles and Components
Derivatives rely on the concept of limits, which allow for precise calculation of rates at a point. Basic rules include the power rule (d/dx [x^n] = n x^{n-1}), product rule, quotient rule, and chain rule for composite functions. These principles enable differentiation of polynomials, trigonometric, exponential, and logarithmic functions.
A Practical Example
Consider the function f(x) = x². Its derivative is f'(x) = 2x. At x = 3, the derivative is 6, meaning the slope of the tangent line at that point is 6. This illustrates how the derivative provides the rate of change; for instance, if x represents time and f(x) distance, f'(x) gives velocity at x = 3.
Importance and Applications
Derivatives are fundamental in analyzing real-world phenomena, such as velocity in physics (derivative of position), marginal cost in economics, and optimization in engineering. They help find maxima and minima, model population growth, and underpin fields like machine learning for gradient descent algorithms.