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The original notation employed by Gottfried Leibniz is used throughout mathematics. It is particularly common when the equation y = f(x) is regarded as a functional relationship between dependent and independent variables y and x. Leibniz's notation makes this relationship explicit by writing the derivative as: [1].
The earliest use of differentials in Leibniz's notebooks may be traced to 1675. He employed this notation in a 1677 letter to Newton. The differential notation also appeared in Leibniz's memoir of 1684. The claim that Leibniz invented the calculus independently of Newton rests on the basis that Leibniz:
Although calculus was independently co-invented by Isaac Newton, most of the notation in modern calculus is from Leibniz. [3] Leibniz's careful attention to his notation makes some believe that "his contribution to calculus was much more influential than Newton's." [4]
Gottfried Wilhelm von Leibniz (1646–1716), German philosopher, mathematician, and namesake of this widely used mathematical notation in calculus.. In calculus, Leibniz's notation, named in honor of the 17th-century German philosopher and mathematician Gottfried Wilhelm Leibniz, uses the symbols dx and dy to represent infinitely small (or infinitesimal) increments of x and y, respectively ...
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:
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.
In Leibniz's notation, this formula is written: =. The reciprocal rule can be derived either from the quotient rule or from the combination of power rule and chain rule. Quotient rule
[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]