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