What is the implicit derivative of #xy+ysqrt(xy-x) =8 #?

1 Answer
Aug 16, 2016

# (dy)/(dx)=(y2sqrt(xy-x)+ y^2-y)/(2xsqrt(xy-x) + 3xy - 2x)#

Explanation:

first lets get the derivatives with respect to #x# and #y#
#d/dx = y+(y(y-1))/(2sqrt(xy-x))#

#d/dx = (y2sqrt(xy-x)+ y(y-1))/(2sqrt(xy-x))#

#d/dy = x+sqrt(xy-x) +xy(1/(2sqrt(xy-x)))#

#d/dy = (2xsqrt(xy-x))/(2sqrt(xy-x))+ (2(xy-x))/(2sqrt(xy-x)) +(xy)/(2sqrt(xy-x))#

#d/dy = (2xsqrt(xy-x)+ 2x(y-1)+(xy))/(2sqrt(xy-x))#

next note that
#d/dx=d/dy⋅dy/dx# or equivalently #d/dy⋅dy/dx -d/dx=0#

which leads to

#(2xsqrt(xy-x)+ 2x(y-1)+(xy))/(2sqrt(xy-x))(dy)/(dx) - (y2sqrt(xy-x)+ y(y-1))/(2sqrt(xy-x))=0#

# (2xsqrt(xy-x)+ 2x(y-1)+(xy))/(2sqrt(xy-x))(dy)/(dx)=(y2sqrt(xy-x)+ y(y-1))/(2sqrt(xy-x))#

# (dy)/(dx)=(y2sqrt(xy-x)+ y(y-1))/(2sqrt(xy-x)) *(2sqrt(xy-x))/(2xsqrt(xy-x) + 2x(y-1)+(xy))#

# (dy)/(dx)=(y2sqrt(xy-x)+ y(y-1))/(2xsqrt(xy-x) + 2x(y-1)+(xy))#

# (dy)/(dx)=(y2sqrt(xy-x)+ y^2-y)/(2xsqrt(xy-x) + 3xy - 2x)#