What is the alternative form of a derivative?
Lim x->a (f(x)-f(a))/x-a
The alternative form of a derivative is known as the differential quotient or a difference quotient.
The differential quotient is an expression of the change in the value of a function over a small change in its independent variable, usually denoted as h.
The differential quotient formula is:
f'(x) = lim (h->0) [f(x+h) – f(x)]/h
This formula represents the slope of the tangent line at a specific point (x, f(x)) on the function f(x). The limit h->0 is the same as the definition of the derivative formula, but this expression provides an alternative form of computing the derivative.
The difference quotient is similar to the differential quotient, but instead of taking the limit as h approaches 0, it considers the change in x across two distinct points.
The difference quotient formula is:
[f(x + h) – f(x)]/[x + h – x]
This expression computes the average rate of change of f(x) over the interval [x, x + h]. As h approaches 0, this average rate of change converges to the true instantaneous rate of change, which is the derivative f'(x) at the point x.
Both the differential quotient and difference quotient formulas can be used to compute derivatives, but they have different applications depending on the context of the problem.
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