Given $x^t * y^m - (x+y)^(m+t)=0$ determine $dy/dx$ ?

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Answer 1 · 2016-10-28T09:00:19

See below.

$f(x,y)=x^t * y^m - (x+y)^(m+t)=0$

$df = f_xdx+f_ydy=0$ so

$dy/dx=-f_x/(f_y)$ but

$f_x=t/x x^ty^m-(t+m)/(x+y)(x+y)^(t+m)$

and

$f_y=m/y x^ty^m-(t+m)/(x+y)(x+y)^(t+m)$

but

$x^ty^m=(x+y)^(t+m)$ then

$f_x=(t/x-(t+m)/(x+y))x^ty^m$ and
$f_y=(m/y-(t+m)/(x+y))x^ty^m$

so

$dy/dx=-(t/x - (t + m)/(x + y))/(m/y - (t + m)/(x + y))=y/x$

Given x^t * y^m - (x+y)^(m+t)=0x^t * y^m - (x+y)^(m+t)=0 determine dy/dxdy/dx ?