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

   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)

   FUNCTION	DERIVATIVE
   $Y=C$	$Y\prime =0$
   $Y=X^n$	$Y\prime =n{X}^{n-1}$
   $Y=c{X}^n$	$Y\prime =cn{X}^{n-1}$
   $Y=\frac{1}{X^n}$	$Y\prime =\frac{-n}{X^{n+1}}$
   $Y=g\left(x\right)+h\left(x\right)$	$Y\prime =g\prime \left(x\right)+h\prime \left(x\right)$
   $Y=g\left(x\right)-h\left(x\right)$	$Y\prime =g\prime \left(x\right)-h\prime \left(x\right)$

   For example,

   $$\begin{array}{c}\text{if }Y=4X^3,\text{then }Y\text{'}=4\left(3\right)X^{3-1}=12X^2\\ \text{if }Y=2X^4,thenY\text{'}=2\left(4\right)X^{4-1}=8X^3\end{array}$$

4. $$\text{if }Y=\frac{1}{X^n},\text{then }Y\text{'}=-n{X}^{-n-1}=\frac{-n}{X^{n+1}}$$ (M6-7)

   Note that $Y=\frac{1}{X^n}$ is the same as $Y=X^{-n}.$ For example,

   $$\begin{array}{c}\text{if }Y=\frac{1}{X^3}\left(\text{or }Y=X^{-3}\right),\text{then }Y\text{'}=-3X^{-3-1}=-3X^{-4}=\frac{-3}{X^4}\\ \text{if }Y=\frac{2}{X^4},\text{then }Y\text{'}=2\left(-4\right)X^{-4-1}=\frac{-8}{X^5}\end{array}$$

5. $$\text{if}Y=g\left(x\right)+h\left(x\right),\text{then}Y\prime =g\prime \left(x\right)+h\prime \left(x\right)$$ (M6-8)

   For example,

   $$\begin{array}{lll}\text{if }Y\hfill & =\hfill & 2X^3+X^2,\text{then }Y\text{'}=2\left(3\right)X^{3-1}+2X^{2-1}=6X^2+2X\hfill \\ \text{if }Y\hfill & =\hfill & 5X^4+3X^2,\text{then }Y\text{'}=5\left(4\right)X^{4-1}+3\left(2\right)X^{2-1}=20X^3+6X\hfill \end{array}$$

6. $$\text{if}Y=g\left(x\right)-h\left(x\right),\text{then}Y\prime =g\prime \left(x\right)-h\prime \left(x\right)$$ (M6-9)

   For example,

   $$\begin{array}{lll}\text{if }Y\hfill & =\hfill & 5X^3-X^2,\text{then }Y\text{'}=5\left(3\right)X^{3-1}-2X^{2-1}=15X^2-2X\hfill \\ \text{if }Y\hfill & =\hfill & 2X^4-4X^2,\text{then }Y\text{'}=2\left(4\right)X^{4-1}-4\left(2\right)X^{2-1}=8X^3-8X\hfill \end{array}$$