alternate version of def. of derivative
The derivative of a function f(x) at a point x=a can be alternatively defined as the instantaneous rate of change of the function at that point
The derivative of a function f(x) at a point x=a can be alternatively defined as the instantaneous rate of change of the function at that point.
To understand this alternate definition, let’s consider a function f(x) and take two points on its graph, (a,f(a)) and (a+h, f(a+h)), where h denotes a small number close to zero. The slope of the line passing through these two points can be calculated using the slope formula:
Slope = (change in y) / (change in x) = (f(a+h) – f(a))/(a+h – a) = (f(a+h) – f(a))/h
As h approaches zero, this slope value becomes the instantaneous rate of change of the function at x=a. In other words, taking the limit as h approaches zero, we can define the derivative of f(x) at x=a as:
f'(a) = lim(h->0) (f(a+h) – f(a))/h
This alternate definition emphasizes the concept of instantaneous rate of change, which can be visualized as the slope of the tangent line to the graph of the function at x=a. It provides a way to calculate the derivative using the concept of limits, showing how the function behaves as the input values get arbitrarily close.
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