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

1 Answer
Oct 28, 2016

See below.

Explanation:

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