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#