Definition of derivative of function at a point.
The derivative of a function at a specific point represents the rate at which the function is changing at that point
The derivative of a function at a specific point represents the rate at which the function is changing at that point. It measures the instantaneous rate of change of the function’s output with respect to its input.
More formally, let’s consider a function f(x) and a specific point a on its domain. If the derivative of f at the point a exists, it is denoted as f'(a) or dy/dx |x=a. The derivative is defined as the limit of the difference quotient as the change in the input approaches zero:
f'(a) = lim(h->0) [f(a+h) – f(a)] / h
Here, h represents a small change in the input variable around the point a. The numerator [f(a+h) – f(a)] measures the change in the function’s output, and dividing it by h provides an approximation of the average rate of change. Taking the limit as h approaches zero gives the instantaneous rate of change at point a.
Geometrically, the derivative corresponds to the slope of the tangent line to the graph of the function at the point (a, f(a)). If the derivative is positive, it indicates an increasing function, while a negative derivative represents a decreasing function. Zero derivative implies a horizontal tangent line, suggesting no change in the function’s value at that point.
The concept of derivatives is fundamental in calculus and has applications in various fields, such as physics, economics, and engineering. It allows us to understand and analyze the behavior of functions, optimize functions, determine the slope of curves, and find rates of change.
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