I want to solve dy/dx for the following: x^2+y^2=R^2 where R is a constant. I know to use implicit differentiation, though I have a question. When I derive R^2, do I obtain 2R or 0? Additionally, deriving y^2 with respect to x yields 2y(dy/dx)? This is different from a partial derivative?
I want to solve for the following:
where is a constant.
I know to use implicit differentiation, though I have a question. When I derive , do I obtain or 0?
Additionally, deriving with respect to x yields ? This is different from a partial derivative?
Answer & Explanation
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 .
By the chaine rule you will get