Mastering Derivatives: The Alternative Definition Of Derivative For Algebraic Calculations

Alternative form of the definition of the derivative

lim x->c. f(x)-f(c) / x-c

The derivative of a function f(x) at x=a can be defined alternatively as:

lim (h->0) [f(a+h) – f(a)] / h

This definition expresses the rate at which the function f(x) changes at the point x=a, by measuring the slope of the tangent line to the curve of f(x) at that point. It is a formalization of the idea that the derivative represents the instantaneous rate of change of the function at a specific point, as the limit as the change in x becomes infinitesimally small.

This definition is equivalent to the more traditional definition of the derivative as:

lim (x->a) [f(x) – f(a)] / (x-a)

Both definitions express the same idea about the derivative, but the alternative form can be easier to use when computing derivatives algebraically, particularly for more complex functions.

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