Development of the concept and deduction of formulae of calculus are outside the purview of our discussion, though only the relevant areas are touched upon, for better conceptualization of the economic tools discussed later in this book.
One of the important concepts in calculus is derivative. To describe derivative of a few important forms assume that y is a variable which is a function of x, i.e., y = ƒ (x).
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1. Derivative:
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Derivative is a measure of change in variable y for a very small change in variable x. usually, the English letter ‘d’ represents small change. Derivatives of a few functional forms are described below:
2. Partial Derivative:
Partial derivatives are defined as derivatives of a function of multiple variables when all but the variable of interest are held fixed during the differentiation. Algebraically, partial derivative of two variables (say, x and y), can be described as derivative of the function with respect to x (or y), holding (or x) constant.
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Suppose a person is interested in prediction of demand for a particular product. There may be various factors that influence its demand like price of the product (P), advertising expenditure (A), income of the prospective customers (I) etc.
Now, if the person wants to measure the rate of change in demand for the product as a result of change in only one variable out of the entire set of independent determinants of demand, he will have to use the tool called partial derivative.
Assume that demand D depends upon three variables P, A and I, i.e., D = ƒ (P, A, I). If the person wants to measure the effect of change in advertising expenditure only, keeping the other variables i.e., A and I, then the rate of change of D with respect to A only (holding P and I constant) is called the partial derivative of D with respect to A and is denoted by ƒA (P, A, I). Similarly, partial derivative of D with respect to P and I are denoted by ƒP (P, A, I) and fI (P, A, I).
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Let f(x, y) be a function of two variables x and y. Then partial derivatives are defined as
