NOTES 3: DIFFERENTIATION RULES Name: Date: Perio: LESSON 3. DERIVATIVE OF POLYNOMIALS AND EXPONENTIAL FUNCTIONS Eample : Prove f ( ) 6 is not ifferentiable at 4. Practice Problems: Fin f '( ) using the efinition.. f ( ) 3. f ( ) 4 3. f ( ) 73 The Constant Rule The erivative of a constant function is 0. That is, if c is a real number, then c 0. Eamples: AP Calculus AB Chapter 3 Notes Page
The Power Rule If n is a rational number, then the function given n by f ( ) is ifferentiable an n n n For f to be ifferentiable at 0 number such that containing 0. n, n must be a is efine on an interval Eamples: The Constant Multiple Rule The Sum an Difference Rules If f is a ifferentiable function an c is a real number, then cf is also ifferentiable an cf ( ) cf '( ) 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 '( ) Eamples: Eamples: Practice Problems: Use the basic ifferentiation rules to fin the erivative of the following problems. 4. f ( ) 5 5. gq ( ) 4 6. q mz () z 4z3 z 7. h 3 ( ) 6 7 8. q ( ) 9. q ( ) p AP Calculus AB Chapter 3 Notes Page
Eample : Given f ( ) e, fin f '( ) using the efinition. Derivative of Natural Eponential Function e e y" y"' Practice Problems: 0. For what values of oes 3 f ( ) 3 6 87 have a horizontal tangent?. At what point on the curve f ( ) e 3 is the tangent line parallel to 3y 5?. Fin an equation of the normal line to the parabola y @ (, 3) 3. Differentiate: y u u 3 u 5 u AP Calculus AB Chapter 3 Notes Page 3
LESSON 3. THE PRODUCT AND QUOTIENT RULES Recap Power Rule: Eponential Rule: Sum/ifference Rule: Constant multiple Rule: Eample: Fin f '( ) f ( ) 3 The Prouct Rule The prouct of two ifferentiable functions f an g is itself ifferentiable. Moreover, the erivative of fg is: f ( ) g ( ) f ( ) g '( ) g ( ) f '( ) Eamples: Fin f '( ) f ( ) 3 Proof of the Prouct Rule The Quotient Rule The quotient of two ifferentiable functions f an g is itself ifferentiable at all values of for which. Moreover, the erivative of f/g is: g ( ) 0 A trick to remember this rule: LowDHigh HighDLow Low f ( ) g( ) f '( ) f ( ) g '( ) g( ) g ( ) AP Calculus AB Chapter 3 Notes Page 4
Practice Problems: Fin the erivative of the following problems.. f ( ) e 3 5. f( ) 6 e 9 a 3. ta ( ) a 4. s( m) m 3 3m m 6m 9 5. Fin the equations of the tangent lines to f that passes through 0, 3 ( ). y AP Calculus AB Chapter 3 Notes Page 5
LESSON 3.3 DERIVATIVES OF TRIGONOMETRIC FUNCTIONS Derivatives of Trigonometric Functions Proofs sin cos tan sec sec sec tan cos sin cot csc csc csc cot Practice Problems: # 4: Fin the erivative.. f ( ) sin cos sec. f( ) 3. f ( ) e cos 4. f( ) sin cos AP Calculus AB Chapter 3 Notes Page 6
Practice Problems: 5. Fin the points on tangent is horizontal. cos f( ) at which the sin sec 6. Given f( ), fin the value(s) of where tan the function has a horizontal tangent. Warm-Up:. Fin f '( ) given cos f( ). Fin f '( ) given f ( ) e tan 3. Given f( ), fin the equations of the tangent lines to f( ) that are parallel to y. 4. Prove: sin D cos AP Calculus AB Chapter 3 Notes Page 7
LESSON 3.4 THE CHAIN RULE The Chain Rule If y f ( u) u g( ) y f ( g( )) y y u an or, equivalently, u ( ( )) f '( ( )) '( ) Derivative of Natural Eponential Function is a ifferentiable function of u an is ifferentiable function of, then is a ifferentiable function of Let u be a ifferential function of.. e e Eamples:. y y 3. 00 y t t 3. 5 4. y 3 3 /5 We shoul use the Chain Rule for these problems. Eamples: Theorem 8.5: Derivatives for Bases other than e e e u u u. ' Let a be a positive real number a a a a u u ln '. ln. a an let u be a ifferential function of. 3. log a ln a u' 4. log a u ln a u Eample: Fin the erivative. Ol methos: f ( ) 7 Chain Rule: f ( ) 7 AP Calculus AB Chapter 3 Notes Page 8
Practice Problems: Fin the erivative.. f ( ) sin 7. f ( ) 3 3. f ( ) sec 4. f ( ) 9 e 5. f ( ) cot e 6. f ( ) 3 6 4 7. 3 f ( ) 5 7 5 8. f( ) 6 3 9. f ( ) cos3 0. f ( ) cos 3 AP Calculus AB Chapter 3 Notes Page 9
Warm-Up: Fin the erivative.. f ( ) tan3. f ( ) sin cos5 3. 3 f( ) 3 4. f ( ) 5. f( ) sin 4 6. If h() 4 an h'() 3, then fin when. h( ) AP Calculus AB Chapter 3 Notes Page 0
LESSON 3.5 IMPLICIT DIFFERENTIATION Functions written in Implicit Form 3 y 5.. y 3. 3 y y 4 Functions written in Eplicit Form.. y y 3 5 3. Not possible to solve for y. Note: To fin the erivative for eample 3, we nee to use IMPLICIT DIFFERENTIATION (with respect to ). Guielines for Implicit Differentiation. Differentiate both sies of the equation with respect to.. Collect all terms involving y on the left sie of the equation an move all other terms to the right sie of the equation. 3. Factor y out of the left sie of the equation. 4. Solve for y by iviing both sies of the equation by the left-han factor that oes not contain y. Eample: Fin the erivative.. 3 y y 4. 3 3 3 4 y y 4 Practice Problems: Fin the erivative.. 3 y y 5y 4. 3 y y AP Calculus AB Chapter 3 Notes Page
3. e y y y 3 4. e y cos 5 sin y Practice Problems: 5. Fin the equation of the tangent line to at y 6. Fin the equation of the tangent line to y y 5 3,. at 7. Determine the slope of the graph of the relation 3 3y y, 3 at 8. Fin y given tan( y) y 6. 9. Fin y given y sin. 0. Fin y given y tan. AP Calculus AB Chapter 3 Notes Page
Derivatives of Inverse Trigonometric Functions MEMORIZE!!! sin tan cos cot sec csc Practice Problems: Fin the erivative.. y tan 5. y sec e Practice Problem 3: Fin y for 4y 8. AP Calculus AB Chapter 3 Notes Page 3
DERIVATIVES OF INVERSE FUNCTIONS Derivative of Inverse Function g f ( ) ( ) g'( ) f f ' ( ) Proof: Eamples: 3. f ( ) an g '(). g f ( ) ( ), fin. f ( ) sin, fin f ( ). g( ) f ( ), fin g '(0) Practice Problems: g( ) f ( ). Fin g'( c ). 3 f ( ) 5 7 4 an g '(4). 5 3 f ( ) 3 an g '(6) Practice Problem 3: f (), 6 f '(), f '() 3, 5 g( ) f ( ), fin g '() AP Calculus AB Chapter 3 Notes Page 4
LESSON 3.6 DERIVATIVES OF LOGARITHMIC FUNCTIONS Review y ln Domain: Range: Asymptote: Prouct: Quotient: Power: Change of Base Formula: Derivative of Natural Logarithmic Function Let u be a ifferential function of... u ln, 0 u u' ln, u 0 u u 3. log a 4. log u a ln a u' ln a u Practice Problems: Fin the erivative.. y ln cos. y ln 3. 9 y ln 9 4. y log 3 e 5. y log sec 6. y 6 7 AP Calculus AB Chapter 3 Notes Page 5
e sin 7. y ln 9 Logarithmic Differentiation To ifferentiate the function y u. Take ln of each sie: ln y lnu, use the following steps:. Epan lnu completely y' y 3. Differentiate implicitly: lnu 4. Solve fore y' y lnu 5. Substitute for y an simplify: y' u lnu Practice Problem 6: Fin f '( ) using Logarithmic Differentiation given: 8. y 9. y (cos ) 0. y 3 5 6 7 cos 3 5e 4 AP Calculus AB Chapter 3 Notes Page 6
LESSON 3.7 RATES OF CHANGE IN THE NATURAL AND SOCIAL SCIENCES Eample : meters. A particle moves along the -ais. Let its position at t secons be given by 3 s( t) t 6t 9t a. Fin the velocity an acceleration functions. Inclue unit. b. When is the particle at rest? c. When is the particle moving left?. Graph the particle s position on 0, 5. e. Fin the total istance travele in these 5 secons. f. When is it speeing up an slowing own? AP Calculus AB Chapter 3 Notes Page 7
Eample : Given 3 s( t) 8t t miles an t is ays. a. Fin the total istance travele on, 4. b. Graph the movement. c. Where is it speeing up? Slowing own? Eample 3: Given 3 f ( ) 5 7. Where is it speeing up? Slowing own? Eample 4: Given P( ) R( ) C( ). The cost to prouce shoes at a factory is $.50/ pair with a $300 start up cost. The owner can sell the shoes at $400/ pair. a. What is the cost, revenue an profit equation for the business? b. What is the marginal cost to make a pair of shoes? c. If the cost function changes to shoes. C( ) 3000 4. 3 5.768 3, fin the marginal cost to make 0. Fin C() C(0) with the new cost function an interpret the results. AP Calculus AB Chapter 3 Notes Page 8
LESSON 3.9 RELATED RATES GOAL STEPS In relate rate problems we compute the rate of change of one quantity in terms of the rate of change of another quantity. We fin an equation that relates two quantities an then use the Chain Rule to ifferentiate both sies with respect to time.. Rea an ientify rates an variables. Ientify equation that relates all rates an variables 3. Take erivative with respect to time using implicit ifferentiation 4. Plug in an solve Practice Problems:. Air is being pumpe into a spherical balloon so that 3 its volume increases at a rate of 00 cm /sec. How fast is the raius of the balloon increasing when the iameter is 50cm?. A laer 5ft long rests against a vertical wall. It slies own the wall at a rate of ft / hr, fin how fast the laer sliing is away from the house when the top of the laer is 7ft high on the wall. Assume the house is perpenicular to the groun. 3. A water tank has the shape of an inverte circular cone with base raius m an height 4m. If the water 3 is being pumpe into the tank at a rate of m / min, fin the rate at which the water level is rising when the water is 3m eep. 4. A triangle is shrinking. Its base ecreases at a rate of 3 cm / min an its height ecreases at a rate of cm / min. Fin how fast the area of the triangle is changing when the triangle has a base of 6cm an a height of 8cm. AP Calculus AB Chapter 3 Notes Page 9
5. An oil spill is epaning an the raius is increasing at a rate of m / sec. Fin how fast the size of the spill is increasing when the iameter is 0m. 6. A rocket is tracke by a raar station. The raar is 5 miles from the launch pa. How fast is the rocket rising when it is 4 miles high an its istance from the raar is increasing at a rate of 000 mi / hr. 7. A plane is flying horizontally at an altitue of 4000ft an at 500 ft /sec above the observation platform. A person on the platform watches the plane fly overhea. How fast is the angle of elevation from the platform to the plane changing when the plane is 5000ft from the platform? 8. Mr. Shay (not Mrs. Nguyen) rinks the bloo of freshman out of a cone shape cup. The cone is tall an has an opening of 8. He fille the cup at a 3 rate of in / min. Fin how fast the height of the bloo is increasing when the total bloo is 4 tall. 9. A 6ft tall woman is walking away from a 5ft tall street light. a. How fast is the length of her shaow changing if she walks at a rate of 3 ft / sec when she is 4ft from the light? b. How fast is the tip of her shaow moving? AP Calculus AB Chapter 3 Notes Page 0
LESSON 3.0 LINEAR APPROXIMATIONS AND DIFFERENTIALS Linear Approimation or Tangent Line Approimation Equation of a tangent line: y f ( a) f '( a)( a) The approimation: f ( ) f ( a) f '( a)( a) Practice Problems: 4 3. f ( ) 3 5 8 Fin the equation of the tangent line @ 3. Fin L (3).. Approimate 3 8. using Linear Approimation. AP Calculus AB Chapter 3 Notes Page