Section 3.6

implicit differentiation - a method of solving for finding both x and the y explicitly using only an implicit formula

An implicit formula: ((x^2)+1)y = (x^2) - 1
An explicit Formula: y = f(x) = ((x^2)-1)/((x^2)+)

You can find dy/dx of an Implicit Differentiation
1. by finding the derivatives of both sides of the equation in respect to x. (Any term involving the varible y must involve the factor dy/dx)
2. Solve the result for dy/dx in terms of x and y.

This method of implicit differentiation is for those equations which have an x and a y on one side of the equation and you cannot solve the equation in respect to x.

a simple notes/book example: y^2 = x

(d/dx)(y^2) = (d/dx)(x)
and since y = f(x)
(d/dx)(y^2) = (d/dx)[f(x)]^2 write y = f(x)
= 2f(x)f'(x) Use the Chain Rule
= 2y(dy/dx) Return to using y instead of f(x)
2y(dy/dx) = 1
(dy/dx) = (1/2y)

Related Rates: x and y are a function of a third variable "t"
Up to this point, we have found that the y is moved by the x, but now both x and y are moved by a third variable "t"

we look for both (dx/dt) and (dy/dt) in these related rates.

a cylinder with radius = 3cm
has water flowing into it at 21 cm^3 per minute

the volume formula of the cylinder is V = 9(pi)(height)

(dV/dt) = 9(pi)(dH/dt)
((21cm^3)/1min) = 9(pi)(dH/dt)
((21cm^3)/(9[pi]min)) = (dH/dt)

If in need of more assistance go HERE

GOGOGOGO Drew Titus next BLOG

Some Jokes in Good Taste
New York (CNN). At John F. Kennedy International Airport today, a Caucasian male (later discovered to be a high school mathematics teacher) was arrested trying to board a flight while in possession of a compass, a protractor and a graphing calculator.
According to law enforcement officials, he is believed to have ties to the Al-Gebra network. He will be charged with carrying weapons of math instruction.

Some engineers are trying to measure the height of a flag pole. They only have a measuring tape and are quite frustrated trying to keep the tape along the pole: It falls down all the time.
A mathematician comes along and asks what they are doing. They explain it to him.
"Well, that's easy..."
He pulls the pole out of the ground, lays it down, and measures it easily.
After he has left, one of the engineers says: "That's so typical of these mathematicians! What we need is the height - and he gives us the length!"