## November 22, 2006

### Scribe Post

Hello everybody, Mark is your scribe for today.

We started off by looking at this equation:

x2+y2=25
This is not a function, because it is continuous.

x2+(f(x))2
The above equation can be re-written like this, where y is some function of x

(f(x)) is the composite of functions and we can differentiate using the Chain rule.
Since we are finding the derivative, We are finding the derivative with respect to x because we rewrote the function (y) in terms of x.

IN GENERAL:
If you use the chain function to find the derivative of a composite, you find the derivative of f with respect to u. Which looks like this

F(x)=f(u(x))

Original Notation
F'(x)= f'(u(x)) x u'(x)

*new notation*
Leibniz Notation
F'(x) = df/du x u'(x)
=
df/du x du/dx
= df/dx

Finding the implicit definition of x2 + y2 = 25

2x + 2yy' = 0

2yy '= -2x

y' = -2x/2y

y' = -x/y
This is the most common way of seeing the Leibniz notation. The x value will be given so all you have to do is plug it in, but to find the y value, we must go back to the original equation to find the y value.

The steps used in the above implicit differentiation requires to think abstract. We must remember y is a composite of functions. Also when a function is given, to write the implicit definition we assume the function can be written in terms of x.

NOTE: When finding the values of y on a function, use the difference of squares. The difference of squares will show the negative and positive values.

Here are questions that we for the last half of the class.

1) find dy/dx @ x=10, for x-2y=4

x - 2y = 4

1 - 2y' = 0

-2y' = -1

y' = 1/2

2)find dy/dx @ x=12, for x - 2y2 = 4

1 - 4y'= 0

1 = 4y'

1/4y' = y' y Dependant

so we go back to original equation and solve for y

12 - 2y2 = 4

8 = 2y2 = 4

4 = y2

y2 - 4 = 0

y= 4, -4

@ (12. 2) y' = 1/8

@ (12,-2) y' = -1/8

3)4x2 + 9y2 = 36
find (1, 4/3 x square root of 2)

8x + 18yy' = 0

9yy' = -4x

y' = -4x/9y

y'
(1, 4/3 x square root of 2) = -4(1) / 9(3 x square root of 2)

= -4/(12 x square root of 2)

= -1/(3
x square root of 2)

4)12x2 - 3y2 = 48

24x - 6yy' = 0

yy' = -4x

y' = -4x/y

12x2 = 48

x2 = 4

x2 - 9 =0

x = 2, -2

Hw: 4.5 all odds, flickr pics, tag as apcalc06 and interesting. See post below

CharLENE is our next scribe.