The Differential Calculus

Calculus is concerned with rates of change and as such, its primary interest is in variables as opposed to constants. Suppose, for example, it is observed that an increase in the amount of money spent on advertising results in an increase in the sales of a firm. The hypothesis is that advertising increases sales, or, symbolically, sales is a function of advertising:3.1<math display='block'> <mrow> <mi>S</mi><mi>A</mi><mi>L</mi><mi>E</mi><mi>S</mi><mo>=</mo><mi>f</mi><mo stretchy='false'>(</mo><mi>A</mi><mi>D</mi><mi>V</mi><mi>E</mi><mi>R</mi><mi>T</mi><mi>I</mi><mi>S</mi><mi>I</mi><mi>N</mi><mi>G</mi><mo stretchy='false'>)</mo></mrow> </math>]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$SALES = f(ADVERTISING)$$ The variable sales (S) ‘depends’ on the variable advertising (A). Altering advertising will bring about some change in sales. Now, let us say that expenditure on advertising (A) changes by some small amount. The small amount is conventionally written with the letter d before the symbol representing the variable. The d therefore means nothing more than a very small part of the variable to which it applies. A small amount of A is written dA and is called a differential ofA. Calculus is based on this idea of a small bit of a variable — a differential of the variable.