Table of Contents Chain Rule Product Rule Quotient Rule Implicit ETA Trig Limits Logarithmic.
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Transcript of Table of Contents Chain Rule Product Rule Quotient Rule Implicit ETA Trig Limits Logarithmic.
Table of Contents
Chain RuleChain Rule Product Product RuleRule
Quotient Quotient RuleRule
ImpliciImplicitt
ETAETA
TrigTrig LimitsLimits
LogarithmiLogarithmicc
Chain RuleChain RuleChain RuleChain RuleF(x) = un
F’(x) = nun-1
*** The derivative of a constant is 0**
Example:
F(x)=x3 + 6xF’(x)= 3x2 +6
Practice ProblemPractice ProblemPractice ProblemPractice ProblemChain Rule
F(x)= x3 + 6x
Practice Problem AnswerPractice Problem AnswerPractice Problem AnswerPractice Problem Answer
F(x)= x3 + 6x
F’(x)= 3x2+6x0
= 3x2+6
Product Rule Product Rule
Example:
y= (4x+1)2 (1-x)3
y’= (4x+1)2(3)(1-x)2 (1)+ (1-x)3(2)(4x+1)(4)
=-3(4x+1)2(1-x)2 + 8(1-x)3(4x+1)
= (4x+1)(1-x)2[(-3)(4x+1)+8(1-x)]
=(4x+1)(1-x)2[(-12x-3)+(8-8x)]
=(4x+1)(1-x)2(5-20x)
=5(4x+1)(1-x)2(1-4x)
Multiplication(F*DS + S*DF)
[(First *Derivative of the Second) + (Second * Derivative of the First)]
Practice Problem
Product Rule
F(x)= (8x+3)(2x-1)2
http://i.ehow.com/images/GlobalPhoto/Articles/5223326/ConfusingEquationsR-main_Full.jpg
F(x)= (8x+3)(2x-1)2
F’(x)= (8x+3)(2)(2x-1)(2)+(2x-1)2(8)
=(8x-4)(8x+3)+(32x-16)= (64x2-8x-12)+(32x-16)=64x2+24x-28=4(16x2+6x-7)
Practice Problem Answer
Quotient RuleQuotient RuleDivision
B*DT – T*DBB2
(Bottom*Derivative of Top) – (Top*Derivative of Bottom)
Bottom y= 2-x 3x+1
Example:
=(3x+1)(-1) – (2-x)(3)
(3x+1)2
=-3x-1-(6-3x)(3x+1)2
_-7_
(3x+1)2
http://www.karlscalculus.org/log_still.gif
Practice Practice ProblemProblem
F(x)= 2 .
(5x+1)3
Practice Problem Practice Problem AnswerAnswer
F(x)= 2 . (5x+1)3 F’(x)= (5x+1)3(0) – 2(3)
(5x+1)2(5) (5x+1)6
= -30(5x+1)2
(5x+1)6
= -30 (5x+1)4
Implicit DifferentiationImplicit Differentiation• What is it?
– the process of finding the derivative of a dependent variable in an implicit function by differentiating each term separately, by expressing the derivative of the dependent variable as a symbol, and by solving the resulting expression for the symbol
• ExampleFind the slope of the circle with equation x2 + y2 = 4 at the point
(0, -2). 2x + 2y () = 0. Rearranging gives: = -2x/2y = At the point x = 0, y = -2, = 0.
EXAMPLES…….EXAMPLES…….
EXAMPLE USING TRIG. OH NO!EXAMPLE USING TRIG. OH NO!
Related Rates using implicit Related Rates using implicit differentiation………differentiation………
– Joey is perched precariously the top of a 10-foot ladder leaning against the back wall of an apartment building (spying on an enemy of his) when it starts to slide down the wall at a rate of 4 ft per minute. Joey's accomplice, Lou, is standing on the ground 6 ft. away from the wall. How fast is the base of the ladder moving when it hits Lou?
That's Pythagoras' Theorem applied to the triangle shown x2 + y2 = 102
Differentiating both sides with respect to t gives2x (dx/dt) + 2y (dy/dt) = 0
Find dx/dt given that dy/dt = -4 at the instant when x = 6
2(6) (dx/dt) + 2y(-4) = 0
We need to figure out side y62 +y2 = 100100-36 = 64 √64 = 8Y = 82(6) (dx/dt) + 2(8)(-4) = 0 12(dx/dt) = 64 dx/dt = 32/6 ft per sec.
Logarithmic DifferentiationLogarithmic Differentiation• Another form of differentiation that makes harder problems,
easier ones. Logarithmic differentiation relies on the chain rule as well as properties of logarithms.
• Simple tips to remember– Multiplication = Addition– Division = Subtraction– Exponents become multipliers– Y = ax = ax lna
• ln(1) = 0• lne = 1• lnex = x• ln(xy)= lnx + lny• ln(x/y) = lnx – lny
LogarithmicLogarithmic
Example:y = 2x lny = xln2 = (x)(0) + ln(2)(1) = yln2 2x ln2
PRACTICE PROBLEM!PRACTICE PROBLEM!
Now you try one…..• y = (x2 +1)x2
ETAETA
exponent, trig, angle
1. first bring exponent in front of problem and copy function
2. take derivative of the trig and copy what is inside parenthesis
3. take derivative of parenthesis
Example: F(x)= sin⁵(cosx)
f’(x)=5sin⁴x(cosx)*cos(cosx)(-sinx)
A few more examplesA few more examples
1)y= sin25x 2)y=cos2x3
Y’=2sin5x*cos5x*5 y’=2cosx3*-sinx3*3x2
Y’=-6x2(cosx3)(sinx3)
http://www.fallingfifth.com/files/comics/calculus.png
TrigonometryTrigonometry• With limits:Lim h 0 sinh=1 lim h 0 1-cosh=0 h h• DerivativesSinu=cosu duCosu=-sinu duTanu=sec2u duSecu=(secu)(tanu) duCscu=-(cscu)(cotu) duCotu=-csc2u du
http://www.cs.utah.edu/~draperg/cartoons/jb/watson.gif
Trig Practice ProblemsTrig Practice Problems
Problems1. Y=3sinx-4cosx
2. Sin2x + cos2x=1
3. y=tan(sinx)
Answers1) y’=3cosx-4(-sinx)Y’=3cosx+4sinx
2) y’=-sinx-1/(1+sinx)2
Y’=-(1+sinx)/(1+sinx)2
Y’=-1/(1+sinx)
3)y’= sec2(sinx)*cosx
LimitsLimits• Definition: f’(x)=lim f(x+h)-f(x)
h 0 hHow to find a limit:1. plug x-value into equation and see if you get a numberExample: Lim x 2 (x^2 -4)/x+2= ((2)^2-4)/2+2= 0L’Hopital’s rule: must be used when x is approaching a # and
you get 0/0Lim x a f(x)/g(x)= 0/0, then Lim x a f’(x)/g’(x)
http://techtalk.blogpico.com/files/2009/01/limit_problem.jpg
Limits cont.Limits cont.
• example: lim x 0 sin3x/sin4x= lim x 0 cos3x(3)/cos4x(4)=3/4
Easier Way: use horizontal asymptotes rule when solving for limits as x infinity
Ex: lim x infinity 2x^4/5x^4= 2/5
http://www.math.lsu.edu/~verrill/teaching/calculus1550/mountain.gif
Limit-practice problemsLimit-practice problemsExample
lim x 0 tanx/xSolution: sec²x/1=
1/cos²x=1Now you try some:• lim x 3 5x² -8x -13
x²-5 Lim x 0 sin(5x)
3x lim x 1 x³-1 (x-1) ² lim x 2 3x²-x-10 x²-4 https://www.muchlearning.org/images/frontpage/Step-By-Step-Calculus-ET-Thumbnail-A.png
Derivative of Natural Log
1/ angle times the derivative of the angleY=ln u
y’=(1/u)(du/dx)
Examples:1)Y=ln(cosx) 2)y=(lnx)3
Y’=1/(cosx)*(-sinx) y’=3(lnx)2*(1/x)
http://www.karlscalculus.org/log_still.gif
FRQ 1971 AB1FRQ 1971 AB1FRQ 1971 AB1FRQ 1971 AB1Let f(x)=ln(x) for all x>0, and let g(x)=x2-4 for all
real x. Let H be the composition of f with g, that is, H(x)=f(g(x)). Let K be the composition of g with f, that is, K(x)=g(f(x)).
e. Find H’(7)
FRQ 1971 AB1 AnswerFRQ 1971 AB1 AnswerFRQ 1971 AB1 AnswerFRQ 1971 AB1 Answere. H= ln(x2-4) H’= 1 (2x) x2-4 = 2x x2-4 = 2(7) (7)2-4 = 14 45
Derivative of e
d/dx eu = eu (du/dx)Copy the function and take derivative of the angle
Examples:1)Y=esinx 2)y=x2ex
Y’=esinx*cosx y’=x2ex+2xex
http://www.intmath.com/Differentiation-transcendental/deriv-ex1.gif
Work CitedWork Cited http://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/trigderivdirectory/TrigDerivatives.html http://www.themathpage.com/acalc/exponential.htm http://people.hofstra.edu/Stefan_Waner/trig/trig3.html http://images.google.com/imgres?imgurl=http://jackieannpatterson.com/wp-content/uploads/
calculus_posted.png&imgrefurl=http://jackieannpatterson.com/tag/calculus/&usg=__jo_t8dSCKUn6-Som7uS25hlLUco=&h=999&w=1707&sz=146&hl=en&start=23&um=1&itbs=1&tbnid=HIr80oNPpBPGBM:&tbnh=88&tbnw=150&prev=/images%3Fq%3Dcalculus%26ndsp%3D20%26hl%3Den%26safe%3Dvss%26sa%3DN%26start%3D20%26um%3D1
http://dragonartz.files.wordpress.com/2009/02/vector-techno-background-10-by-dragonart.png?w=495&h=495 http://carlasenecal.com/portfoliosite2/images/background2.gif http://img1.visualizeus.com/thumbs/09/02/03/backgrounds,color,graphic,design,light,pink,purple-97fbd6173a3d
17d06e689e9f8980d86a_h.jpg http://www.wisegorilla.com/images/backgrounds/math.jpg http://www.wallcoo.net/holiday/Christmas_illustration_07_vladstudio/images/
Christmas_wallpaper_sparks.jpg http://www.karabudd.com/Images/Background.jpg http://www.psdgraphics.com/wp-content/uploads/2009/06/flow-background.jpg http://maurergraphics.com/images/background.gif http://www.gaialandscapedesignbackgrounds.net/landscape-design-background--zen-Hong-Kong-nochoice.jpg http://www.webpagebackground.com/designs/alienskin.jpg http://tygrp.moo.jp/blog/summer_20070604-thumb.jpg http://bluemist.com/imgs/thebackground.jpg
© Andrea Alonso, Emily Olyarchuk, Deana Tourigny February 19, 2010
Table of Contents
1. Chain Rule2. Product Rule3. Quotient Rule4. Implicit5. Logarithmic6. ETA7. Trig8. Limits