Suppose is a function of and ; then
if we define
we may also write (1) as
by the implicit funtion theorem, this equation in fact may be seen as defining , a function of , provided that
under such circumstances, we may affirm is uniquely determined as a differentiable function of in some neighborhood of any point ; then we have
we may take the total derivative with respect to x to obtain
a little algebra allows us to isolate the terms containing :
for the sake of compactess and brevity, we introduce the subscript notation
and write (11) in the form
which gives a general expression for ; in the event that is constant, we obtain
which the reader may recognize as the slope of the circle
at any point where .
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