Alternative form of the definition of the derivative
lim x->c. f(x)-f(c) / x-c
One alternative form of the definition of the derivative of a function is:
The derivative of a function f at a point x is the limit as h approaches 0 of the difference quotient (f(x + h) – f(x))/h, if this limit exists.
In other words, the derivative of a function at a given point x can be thought of as the slope of the tangent line to the function at that point. It is the rate of change of the function with respect to its input variable, x.
This alternative form of the definition of the derivative is commonly used in conjunction with the formal definition involving limits. It provides a more intuitive understanding of what the derivative represents and how it can be used to analyze the behavior of functions. Additionally, this form of the definition can be used to calculate the derivative of a function using the limit laws of calculus, without having to resort to the formal definition every time.
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