THEOREM: THE CONSTANT RULE

Transcription

1 MATH /MYERS/ALL FORMULAS ON THIS REVIEW MUST BE MEMORIZED! DERIVATIVE REVIEW THEOREM: THE CONSTANT RULE The erivative of a constant function is zero. That is, if c is a real number, then c 0 Eample 1: Fin the erivative of the function g 5. THEOREM: THE POWER RULE If n is a rational number, then the function f n is ifferentiable an n n n1 For f to be ifferentiable at 0, n must be a number such that efine on an interval containing zero. Eample : Fin the following erivatives. a. f 5 b. f 1 n 1 is THEOREM: THE CONSTANT MULTIPLE RULE If f is a ifferentiable function an c is a real number, then cf is also ifferentiable an cf cf Eample 3: Fin the slope of the graph of f 4 3 at a. b. 15 c. 64 pg. 1

2 THEOREM: THE SUM AND DIFFERENCE RULES The sum (or ifference) of two ifferentiable functions f an g is itself ifferentiable. Moreover, the erivative of f g (or f g ) is the sum (or ifference) of the erivatives of f an g. f g f g f g f g Eample 4: Fin the equation of the line tangent to the graph of at = 4. f 1. Fin at.. Fin slope at. 3. Fin the of the line to the graph at using the - form of the equation of the line. pg.

3 THEOREM: DERIVATIVES OF THE TRIGONOMETRIC FUNCTIONS sin cos cos sin csc csc cot sec sec tan tan sec cot csc Eample 5: Fin the erivative of the following functions: f a. sin 6 b. r 5 3cos c. ht cos t cot t. f 1 7sec e. f tan pg. 3

4 THEOREM: THE PRODUCT RULE The prouct of two ifferentiable functions f an g is itself ifferentiable. Moreover, the erivative of fg is the erivative of the first function times the secon function, plus the first function times the erivative of the secon function. f g f g f g This rule etens to cover proucts of more than two factors. For eample the erivative of the prouct of functions fghk is fghk f g h k f g h k f g h k f g h k Eample 6: Fin the erivative of the following functions. Simplify your result to a single rational epression with positive eponents. g a. cos b. ht 3 t pg. 4

5 THEOREM: THE QUOTIENT RULE The quotient of two ifferentiable functions f an g is itself ifferentiable at all values of for which g 0. Moreover, the erivative of f g is the erivative of the numerator times the enominator, minus the numerator times the erivative of the enominator, all ivie by the square of the enominator. f f g f g g g Eample 7: Fin the erivative of the following functions. Simplify your result to a single rational epression with positive eponents. a. g 5 3 b. ht t t 1 c. ht cot t t pg. 5

6 Eample 8: Fin the given higher-orer erivative. a. f, f b. 4 6 f 1, f Theorem: The Chain Rule If y f u is a ifferentiable function of u an u g y f g function of, then is a ifferentiable function of an y y u or f g f g g. u is a ifferentiable Eample 9: Fin the erivative using the Chain Rule. a. y 10 b. f 1 pg. 6

7 Eample 10: Fin the erivative of the following functions. a. y sin b. y sin c. y sin. y sin e. y cos f. f 3 pg. 7

8 h sin 4 g. Theorem: Derivative of the Natural Logarithmic Function Let u be a ifferentiable function of. 1 u 1. ln. ln u, u 0 u Eample 11: Fin the erivative. a. y ln 3 b. f ln cos h t c. ln pg. 8

9 Theorem: Derivative of the Natural Eponential Function Let u be a ifferentiable function of. u u 1. e e. e e u Eample 1: Fin the erivative. a. y e sin b. f e c. ht t e ln e t pg. 9

10 Theorem: Derivatives for Bases other than e Let a be a positive real number a 1 an let u be a ifferentiable function of. u u 1. a ln aa. a ln aa u 1 u 3. log a 4. loga u ln a ln a u Eample 13: Fin the erivative. a. 3 y b. f log5 THEOREM: DERIVATIVES OF THE INVERSE TRIGONOMETRIC FUNCTIONS (u is a function of ) u u arcsin u arccos u 1 u 1 u u u arccsc u arcsec u u u 1 u u 1 u u arctan u arccot u 1 u 1 u pg. 10

11 Eample 14: Fin the erivative. a. y arctan 3 ln 1 9 b. f arcsin pg. 11