Given y = x/(x+y), find dy/dx.
If you differentiate both sides wrt* x, using the quotient rule on the right, you get:
dy/dx = [(x+y)(1) - (1 + dy/dx)(x)]/(x+y)[sup]2[/sup]
which ends up simplifying to
dy/dx = y/[(x+y)[sup]2[/sup]+x]
However, if you multiply both sides by (x+y)…
y(x+y) = x
and THEN differentiate (using the product rule on the left) and simplify, you get
dy/dx = (1-y)/(2y+x)
As far as everyone in my class (including the teacher) can tell, these two answers are not equal, so one of them must be wrong. Which one, and why?
*with respect to. I learned that in another calculus thread a few days ago.
TJdude825:
Given y = x/(x+y), find dy/dx.
If you differentiate both sides wrt* x, using the quotient rule on the right, you get:
dy/dx = [(x+y)(1) - (1 + dy/dx)(x)]/(x+y)[sup]2[/sup]
which ends up simplifying to
dy/dx = y/[(x+y)[sup]2[/sup]+x]
However, if you multiply both sides by (x+y)…
y(x+y) = x
and THEN differentiate (using the product rule on the left) and simplify, you get
dy/dx = (1-y)/(2y+x)
As far as everyone in my class (including the teacher) can tell, these two answers are not equal, so one of them must be wrong. Which one, and why?
*with respect to. I learned that in another calculus thread a few days ago.
Actually, they are the same.
(2) y’=(1-y)/(2y+x) substitute y = x/(x+y) into the numerator to get
I separated out the term in square brakets, because, using y = x/(x+y) can be rearranged to y^2+xy=x . Sub this into (3) and you have the same answer as your first method.
Balduran:
Actually, they are the same.
(2) y’=(1-y)/(2y+x) substitute y = x/(x+y) into the numerator to get
I separated out the term in square brakets, because, using y = x/(x+y) can be rearranged to y^2+xy=x . Sub this into (3) and you have the same answer as your first method.
That’s it. More succinctly (in case anyone balked at the formulas), the OP’s class forgot that when doing algebraic manipulations to compare the two answers they’re allowed (in fact, required) to use the original relation between x and y.