Differentiation

Finishing This Unit

Review the objectives of this unit and make sure you are able to meet all of them. In particular you should be able to
- differentiate functions with ease.
- differentiate trigonometric functions.
- apply all the rules of differentiation.
- identify the different notations of the derivative.
- interpret the derivative in two different ways.
If there is a concept, definition, example or exercise that is not yet clear to you, go back and reread it, then contact your tutor for help.
Tables of differentiation rules are presented below. You may wish to print these diagrams and pin them above your desk for easy reference.
Do the exercises in “Learning from Mistakes” section for this unit.
You may want to do Exercises 1-4, 13-42, 45-47, 69, 71, 72, 74, 76, 81(a) and 83 from the “Review” (pages 203-207 of the textbook).

Name		Rule
Power Rule		$\frac{d}{d x} x^{r} = r x^{r - 1}$
Reciprocal Rule		$\frac{d}{d x} \frac{1}{g (x)} = - \frac{g^{'} (x)}{g {(x)}^{2}}$
Product Rule		$\frac{d}{d x} f (x) g (x) = f^{'} (x) g (x) + f (x) g^{'} (x)$
Quotient Rule		$\frac{d}{d x} \frac{f (x)}{g (x)} = \frac{f^{'} (x) g (x) - f (x) g^{'} (x)}{g {(x)}^{2}}$
Chain Rule		$\frac{d}{d x} f (g (x)) = f^{'} (g (x)) g^{'} (x)$
General Power Rule		$\frac{d}{d x} g {(x)}^{r} = r g {(x)}^{r - 1} g^{'} (x)$

$r$		$\frac{d}{d x} x^{r} = r x^{r - 1}$
$r = - 1$		$\frac{d}{d x} \frac{1}{x} = - \frac{1}{x^{2}}$
$r = \frac{1}{2}$		$\frac{d}{d x} \sqrt{x} = \frac{1}{2 \sqrt{x}}$
$r = - \frac{1}{2}$		$\frac{d}{d x} \frac{1}{\sqrt{x}} = - \frac{1}{2 x^{3 ∕ 2}} = - \frac{1}{2 \sqrt{x^{3}}}$

$r$		$\frac{d}{d x} g {(x)}^{r} = r g {(x)}^{r - 1} g^{'} (x)$
$r = - 1$		$\frac{d}{d x} \frac{1}{g (x)} = - \frac{g^{'} (x)}{g {(x)}^{2}}$
$r = \frac{1}{2}$		$\frac{d}{d x} \sqrt{g (x)} = \frac{g^{'} (x)}{2 \sqrt{g (x)}}$
$r = - \frac{1}{2}$		$\frac{d}{d x} \frac{1}{\sqrt{g (x)}} = - \frac{g^{'} (x)}{2 g {(x)}^{3 ∕ 2}} = - \frac{g^{'} (x)}{2 \sqrt{g {(x)}^{3}}}$