Limit Definition of Derivative
limit (as h approaches 0)= F(x+h)-F(x)/h
The limit definition of the derivative is a mathematical formula used to compute the instantaneous rate of change of a function at a certain point. This formula is based on the concept of a limit and involves taking the limit as the input variable (often denoted by ‘x’) approaches a certain value (often denoted by ‘a’).
The formula for the limit definition of the derivative is as follows:
f'(a) = lim (h -> 0) [f(a+h)-f(a)]/h
Here, f is the function, f'(a) denotes the derivative of f at the point a, and h is a small increment in the input variable (i.e., h represents the difference between a and the neighboring point).
To use the formula, we first evaluate the expression [f(a+h)-f(a)]/h, which represents the average rate of change of the function over a small interval between a and a+h. Then, we take the limit of this expression as h approaches 0, which gives us the instantaneous rate of change of the function at the point a.
In simpler terms, the limit definition of the derivative essentially calculates the slope of the tangent line to the graph of a function at a specific point. This slope is synonymous with the instantaneous rate of change of the function at that point.
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