How do I use mathematica to implicitly solve differential equations of a single variable?

Question

I'm trying to force Mathematica to implicitly differentiate an ellipse equation of the form:

x^2/a^2+y^2/b^2 == 100

with a = 8 and b = 6.

The command I'm using looks like this:

D[x^2/a^2 + y^2/b^2 == 100/. y -> 3/4*Sqrt[6400-x^2], x]

where, y->3/4*Sqrt[6400-x^2] comes from solving y in terms of x.

I got this far by following the advice found here: http://www.hostsrv.com/webmaa/app1/MSP/webm1010/implicit

Input for this script is the conventional way that an implicit relationship beween x and y is expressed in calculus textbooks. In Mathematica you need to make this relationship explicit by using y[x] in place of y. This is done automatically in the script by replacing all occurances of y with y[x].

But the solution Mathematica gives does not have y' or dy/dx in it (like when I solved it by hand). So I don't think it's been solved correctly. Any idea on what command would get the program to solve an implicit differential? Thanks.