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In mathematics, a rate is the quotient of two quantities, often represented as a fraction. [1] If the divisor (or fraction denominator) in the rate is equal to one expressed as a single unit, and if it is assumed that this quantity can be changed systematically (i.e., is an independent variable), then the dividend (the fraction numerator) of the rate expresses the corresponding rate of change ...
Construct an equation relating the quantities whose rates of change are known to the quantity whose rate of change is to be found. Differentiate both sides of the equation with respect to time (or other rate of change). Often, the chain rule is employed at this step. Substitute the known rates of change and the known quantities into the equation.
The growth rate of output is the time derivative of the flow of output divided by output itself. The growth rate of the labor force is the time derivative of the labor force divided by the labor force itself. And sometimes there appears a time derivative of a variable which, unlike the examples above, is not measured in units of currency:
Jerk (also known as Jolt) is the rate of change of an object's acceleration over time. It is a vector quantity (having both magnitude and direction). Jerk is most commonly denoted by the symbol j and expressed in m/s 3 ( SI units ) or standard gravities per second ( g 0 /s).
The derivative of a function at a chosen input value describes the rate of change of the function near that input value. The process of finding a derivative is called differentiation . Geometrically, the derivative at a point is the slope of the tangent line to the graph of the function at that point, provided that the derivative exists and is ...
[5] [6] The difference quotient is a measure of the average rate of change of the function over an interval (in this case, an interval of length h). [7] [8]: 237 [9] The limit of the difference quotient (i.e., the derivative) is thus the instantaneous rate of change. [9]
A natural question to ask, given the somewhat abstract setting of the general framework above, is whether the rate function is unique. This turns out to be the case: given a sequence of probability measures (μ δ) δ>0 on X satisfying the large deviation principle for two rate functions I and J, it follows that I(x) = J(x) for all x ∈ X.
A linear function () = + has a constant rate of change equal to its slope a, so its derivative is the constant function ′ =. The fundamental idea of differential calculus is that any smooth function f ( x ) {\displaystyle f(x)} (not necessarily linear) can be closely approximated near a given point x = c {\displaystyle x=c} by a unique linear ...