Implicit vs Partial Differentiation

[Jason76]

In some ways similar and other ways not. What is the major difference? Partial Differentiation involves taking the derivative of one variable and leaving the other constant. On the other hand, all variables are differentiated in implicit differentiation. Is this right?

 

[tkhunny]

Given f(x,y), one must assume that y = g(x) or x = h(y) in order to perform Implicit Differentiation. In other words, one must be a function of the other. If they are totally independent, Implicit Differentiation is not appropriate.

 

[HallsofIvy]

With implcit differentiation, you have some f(x,y)= constant, so that y is some (implied) function of x, and want to find dy/dx from that statement. With partial differentiation, you have z= f(x,y), where x and y are independent, and want to find \(\displaystyle \partial f/\partial x\) and \(\displaystyle \partial f/\partial y\). Those are completely different concepts.

There is however, a connection. If we have, say, a "contour" plot of z= f(x,y), showing curves of constant z, then, on one such curve, we have f(x,y)= constant and can say that y must depend on x in order to give this restriction.

 

[Jason76]

With implicit differentiation, both variables are differentiated, but at the end of the problem, one variable is isolated (without any number being connected to it) on one side. On the other hand, with partial differentiation, one variable is differentiated, but the other is held constant.