Formal definition of derivative
limit as h approaches 0 of [f(a+h)-f(a)]/h
The formal definition of a derivative is the limit of the ratio of the change in the output of a function to the change in its input as the change in input approaches zero. Mathematically, this can be represented as:
f'(x) = lim (h -> 0) [(f(x+h) – f(x))/h]
where f'(x) is the derivative of the function f at x, h represents the change in input, and the limit as h approaches 0 ensures that the change in input becomes infinitesimally small.
This definition essentially measures the rate of change of a function at a specific point, showing how quickly the function is changing around that point. By taking the limit as the change in input approaches zero, we can obtain an instantaneous rate of change, or the slope of a tangent line to the function at that point.
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