#tan(x+y)=x#

Regard #y# as a function of x and differentiate with respect to #x#.

#implies (d(tan(x+y)))/dx=(d(x))/dx #

using chain rule

#implies sec^2(x+y)((d(x))/(dx)+(d(y))/(dx))=1#

#implies sec^2(x+y)(1+(d(y))/(dx))=1#

#implies (d(y))/(dx)=1/sec^2(x+y)#

#implies (d(y))/(dx)=1/sec^2(x+y)-1#

#implies (d(y))/(dx)=(1-sec^2(x+y))/sec^2(x+y)#

#implies (d(y))/(dx)=(1-sec^2(x+y))(cos^2(x+y))##...........(i)#

#implies (d(x))/(dy)=1/((1-sec^2(x+y))(cos^2(x+y)))##...............(ii)#

#(i)# represents the derivative of #y# with respect to #x# and #(ii)# represents the derivative of #x# with respect to #y#.