Quantitative Analysis for Management, 13/e

M6.3 Some Common Derivatives

Although derivatives exist for a number of different functional forms, we restrict our discussion to the six most common ones. The references at the end of this module provide additional information on derivatives. Table M6.1 gives a summary of the common derivatives. We provide examples of each of these.

1. $$\begin{array}{ll}\text{If }Y=\hfill & c,\text{ then }Y\text{'}=0\hfill \\ \hfill c& =\text{ constant}\hfill \end{array}$$

   The derivative of a constant is 0.

   For example, if $Y=4,$ then $Y\prime =0.$ The graph of Y is a horizontal line, so the change in Y is zero, regardless of the value of X.

2. $$\text{If}Y=X^n,\text{then}Y\prime =n{X}^{n-1}$$ (M6-5)

   For example,

   $$\begin{array}{c}\text{if }Y\text{ = }{X}^{\text{2}}\text{, then }Y\text{'}\text{ = 2}{X}^{\text{2-1}}\text{ = 2}X\\ \text{if }Y\text{ = }{X}^{\text{3}}\text{, then }Y\text{'}\text{ = 3}{X}^{\text{3-1}}\text{ = 3}{X}^{\text{2}}\\ \text{if }Y\text{ = }{X}^{\text{9}}\text{, then }Y\text{'}\text{ = 9}{X}^{\text{9-1}}\text{ = 9}{X}^{\text{8}}\end{array}$$
3. $$\text{IF}Y=c{X}^n,\text{then }Y\text{'}=cn{X}^{n-1}$$ (M6-6)

   Table M6.1 Some Common Derivatives

   FUNCTION ...