1. By: Andrea Alonso, Emily Olyarchuk, Deana Tourigny
DIFFERENTIATION
By:
Andrea Alonso,
Emily Olyarchuk,
Deana Tourigny
AP CALCULUS
5TH PERIOD

2. Table of Contents Chain Rule Product Rule Implicit Quotient Rule ETA
Logarithmic
ETA
Trig
Limits

3. *** The derivative of a constant is 0**
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

4. Practice Problem
Chain Rule
F(x)= x3 + 6x

5. Practice Problem Answer
F(x)= x3 + 6x F’(x)= 3x2+6x0 = 3x2+6

6. Product Rule Multiplication (F*DS + S*DF) Example:
[(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)

7. Practice Problem F(x)= (8x+3)(2x-1)2 Product Rule

8. 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)

9. (Bottom*Derivative of Top) – (Top*Derivative of Bottom)
Quotient Rule
Division
B*DT – T*DB B2
(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

10. Practice Problem
F(x)= 2 . (5x+1)3

11. 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 = -30 (5x+1)4

12. 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.

13. EXAMPLES…….

14. EXAMPLE USING TRIG. OH NO!

15. 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?

16. 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
= 64
√64 = 8
Y = 8
2(6) (dx/dt) + 2(8)(-4) = 0
12(dx/dt) = 64
dx/dt = 32/6 ft per sec.

17. 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
Y = ax = ax lna
ln(1) = 0
lne = 1
lnex = x
ln(xy)= lnx + lny
ln(x/y) = lnx – lny

18. Logarithmic
Example: y = 2x lny = xln2 = (x)(0) + ln(2)(1) = yln2 2x ln2

19. PRACTICE PROBLEM!

20. Now you try one…..
y = (x2 +1)x2

21. ETA exponent, trig, angle
first bring exponent in front of problem and copy function
take derivative of the trig and copy what is inside parenthesis
take derivative of parenthesis
Example:
F(x)= sin⁵(cosx)
f’(x)=5sin⁴x(cosx)*cos(cosx)(-sinx)
Just explain to class

22. A few more examples y= sin25x 2)y=cos2x3
Y’=2sin5x*cos5x*5 y’=2cosx3*-sinx3*3x2
Y’=-6x2(cosx3)(sinx3)
Walk through examples. I thought the cartoon was funny, so I decided to add it here cause I could find one on ETA.

23. Trigonometry With limits: Lim h 0 sinh=1 lim h 0 1-cosh=0 h h
Derivatives
Sinu=cosu du
Cosu=-sinu du
Tanu=sec2u du
Secu=(secu)(tanu) du
Cscu=-(cscu)(cotu) du
Cotu=-csc2u du
Just explain it, talk about what is exemplified in cartoon

24. Trig Practice Problems
Answers
Y=3sinx-4cosx
Sin2x + cos2x=1
y=tan(sinx)
1) 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

25. Limits Definition: f’(x)=lim f(x+h)-f(x) h 0 h How to find a limit:
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)
Explain and talk about picture

26. 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

27. Limit-practice problems
Example
lim x 0 tanx/x
Solution: sec²x/1=
1/cos²x=1
Now you try some:
lim x x² -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

28. 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) 2)y=(lnx)3 Y’=1/(cosx)*(-sinx) y’=3(lnx)2*(1/x)

29. 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)

30. FRQ 1971 AB1 Answer
e. H= ln(x2-4) H’= 1 (2x) x2-4 = 2x = 2(7) (7)2-4 = 14 45

31. Copy the function and take derivative of the angle
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
Explain slide and graph. Because we know the derivative is another word for slope. The derivative of e is the same as the function, therefore at any given point on this graph the y-value and the slop are the same.

32. Work Cited

33. © Andrea Alonso, Emily Olyarchuk, Deana Tourigny February 19, 2010

34. Table of Contents Chain Rule Product Rule Quotient Rule Implicit
Logarithmic
ETA
Trig
Limits