Table of Contents
Chain Rule
Quotient Rule
Product Rule
Implicit
Logarithmic
ETA
Trig
Limits
Chain Rule
F(x) = un
F’(x) = nun-1
*** The derivative of a constant is 0**
Example:
F(x)=x3 + 6x
F’(x)= 3x2 +6
Practice Problem
Chain Rule
F(x)=
3
x +
6x
Practice Problem Answer
F(x)= x3 + 6x
F’(x)= 3x2+6x0
= 3x2+6
Product Rule
Multiplication
(F*DS + S*DF)
[(First *Derivative of the Second) + (Second * Derivative of the First)]
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)
Practice Problem
Product Rule
F(x)= (8x+3)(2x-1)2
http://i.ehow.com/images/GlobalPhoto/Articles/5223326/ConfusingEquationsR-main_Full.jpg
Practice Problem Answer
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)
Quotient Rule
Division
B*DT – T*DB
B2
http://www.karlscalculus.org/log_still.gif
(Bottom*Derivative of Top) – (Top*Derivative of Bottom)
Bottom
Example: y= 2-x
3x+1
=(3x+1)(-1) – (2-x)(3)
(3x+1)2
=-3x-1-(6-3x)
(3x+1)2
=-3x-1+3x-6
(3x+1)2
_-7_
(3x+1)2
Practice Problem
F(x)=
2
(5x+1)3
.
Practice Problem Answer
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 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
• Example
Find 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…….
EXAMPLE USING TRIG. OH NO!
Related Rates using implicit
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 gives
2x (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 y
62 +y2 = 100
100-36 = 64
√64 = 8
Y=8
2(6) (dx/dt) + 2(8)(-4) = 0
12(dx/dt) = 64
dx/dt = 32/6 ft per sec.
Logarithmic 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
= ax lna
Y = ax
ln(1) = 0
lne = 1
lnex = x
ln(xy)= lnx + lny
ln(x/y) = lnx – lny
Logarithmic
Example:
y = 2x
lny = xln2
(x)(0) + ln(2)(1)
=
= yln2
2x ln2
PRACTICE PROBLEM!
Now you try one…..
• y = (x2 +1)x2
ETA
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 examples
1) y= sin25x
Y’=2sin5x*cos5x*5
http://www.fallingfifth.com/files/comics/calculus.png
2)y=cos2x3
y’=2cosx3*-sinx3*3x2
Y’=-6x2(cosx3)(sinx3)
Trigonometry
• With limits:
Lim h 0 sinh=1
h
• Derivatives
Sinu=cosu du
Cosu=-sinu du
Tanu=sec2u du
Secu=(secu)(tanu) du
Cscu=-(cscu)(cotu) du
Cotu=-csc2u du
lim h 0 1-cosh=0
h
http://www.cs.utah.edu/~draperg/cartoons/jb/watson.gif
Trig Practice Problems
Problems
1. Y=3sinx-4cosx
Answers
1) y’=3cosx-4(-sinx)
Y’=3cosx+4sinx
2. Sin2x + cos2x=1
2) y’=-sinx-1/(1+sinx)2
Y’=-(1+sinx)/(1+sinx)2
Y’=-1/(1+sinx)
3. y=tan(sinx)
3)y’= sec2(sinx)*cosx
Limits
• Definition: f’(x)=lim f(x+h)-f(x)
h 0
h
How to find a limit:
1. plug x-value into equation and see if you get a number
Example: Lim x 2 (x^2 -4)/x+2=
((2)^2-4)/2+2= 0
L’Hopital’s rule: must be used when x is approaching a # and
you get 0/0
Lim 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.
•
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 problems
Example
lim x 0 tanx/x
Solution: sec²x/1=
1/cos²x=1
Now 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 angle
Y=ln u
y’=(1/u)(du/dx)
Examples:
1)Y=ln(cosx)
Y’=1/(cosx)*(-sinx)
http://www.karlscalculus.org/log_still.gif
2)y=(lnx)3
y’=3(lnx)2*(1/x)
FRQ 1971 AB1
Let 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 Answer
e. 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
Y’=esinx*cosx
2)y=x2ex
y’=x2ex+2xex
http://www.intmath.com/Differentiation-transcendental/deriv-ex1.gif
Work Cited
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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/wpcontent/uploads/calculus_posted.png&imgrefurl=http://jackieannpatterson.com/tag/calculus/&usg=__jo_t8dS
CKUn6Som7uS25hlLUco=&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%26sta
rt%3D20%26um%3D1
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http://www.wallcoo.net/holiday/Christmas_illustration_07_vladstudio/images/Christmas_wallpaper_sparks.jp
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http://www.karabudd.com/Images/Background.jpg
http://www.psdgraphics.com/wp-content/uploads/2009/06/flow-background.jpg
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http://www.gaialandscapedesignbackgrounds.net/landscape-design-background--zen-Hong-Kong-nochoice.jpg
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http://bluemist.com/imgs/thebackground.jpg
© Andrea Alonso, Emily Olyarchuk,
Deana Tourigny February 19, 2010
Table of Contents
1.
2.
3.
4.
5.
6.
7.
8.
Chain Rule
Product Rule
Quotient Rule
Implicit
Logarithmic
ETA
Trig
Limits