What is the difference between implicit differentiation and partial derivatives?

Question

and is this true: "partial derivatives"="partial differentiation"?

Answer 1

Partial differentiation is applied to multivariate functions in which the variable to be differentiated against is explicitly specified, and the differentiation done as if all the other variables are constants. Therefore, the partial derivative of y = x^2*z^2 with respect to x would be 2x*y^2. This function of y is explicit: y appears by itself on one side, all other variables are separate. Some funcitions are implicit, meaning the function itself (y) appears in an equation mixed up with the other variables. Sometimes it is difficult to separate the y out, or when you do you end up with a complex function. In that case implicit differentiation is used. This is better explained in the following article:
http://en.wikipedia.org/wiki/Implicit_differentiation

Yes partial derivatives are the result of partial differentiation.

Answer 2

Implicit Differentiation Partial Derivatives

Answer 3

Implicit Partial Differentiation

Answer 4

Let's take a general case, an equation in two variables(for your convenience),

2x^2 + 8x.y^3 = e^y + x^3
differentiate with respect to x, then

4x + 8y^3 + 8x .3y^2 .(dy/dx) = e^y . (dy/dx) + 3x^2
(24xy^2 - e^y) (dy/dx) = 3x^2 - 4x - 8y^3

and,
dy/dx = (8y^3 - 3x^2 + 4x) / (e^y - 24xy^2)

the above is an example of implicit differentiation.
As you might observe it is a pure consequence of the chain rule of differentiation and gives the derivative in terms of both variables.

As for partial differentiation,
z = 2x^2 + 8x.y^3
would have a partial derivative by differentiation with respect to x of,
&z / &x = 4x + 8y^3
,(where & is for the partial differential notation)
As you can see, the result does not rely upon the chain rule and instead finds the derivatives wrt x, treating ANY other term as if they were constant.

There are numerous applications, for example, partial derivatives are necessary to see whether the Cauchy-Riemann conditions are satisfied for a function in complex analysis.

Hope this helps!