formal version of def. of derivative
The formal definition of the derivative of a function f(x) at a point x=a is as follows:
If the limit of the difference quotient as h approaches zero exists, then the derivative of f(x) at x=a is defined as:
f'(a) = lim(h→0) [(f(a + h) – f(a))/h]
In this definition, f'(a) represents the derivative of f(x) at x=a, and h is a small change in the value of x
The formal definition of the derivative of a function f(x) at a point x=a is as follows:
If the limit of the difference quotient as h approaches zero exists, then the derivative of f(x) at x=a is defined as:
f'(a) = lim(h→0) [(f(a + h) – f(a))/h]
In this definition, f'(a) represents the derivative of f(x) at x=a, and h is a small change in the value of x.
The difference quotient (f(a + h) – f(a))/h represents the average rate of change of f(x) over the interval [a, a+h]. Taking the limit as h approaches zero allows us to determine the instantaneous rate of change, or the slope of the tangent line to the graph of f(x) at x=a.
By evaluating this limit, we obtain the derivative of f(x) at x=a, which represents the value of the instantaneous rate of change, or the slope of the tangent line, at that specific point.
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