Given: Implicitly differentiate #2=e^(x-y)/tan(y)#
Change #1/tan(y) to cot(y)#:
#2=e^(x-y)cot(y)#
Differentiate both sides:
#(d(2))/dx= (d(e^(x-y)cot(y)))/dx#
The left side is 0:
#0= (d(e^(x-y)cot(y)))/dx#
Use the product rule on the right side:
#0= (d(e^(x-y)))/dxcot(y)+ e^(x-y)(d(cot(y)))/dx#
Substitute #-csc^2(y)dy/dx# for #(d(cot(y)))/dx#
#0= (d(e^(x-y)))/dxcot(y)+ e^(x-y)(-csc^2(y)dy/dx)#
Write #e^(x-y)# as #e^xe^-y#:
#0= (d(e^xe^-y))/dxcot(y)+ e^(x-y)(-csc^2(y)dy/dx)#
Use the product rule:
#0= (e^xe^-y-e^xe^-ydy/dx)cot(y)+ e^(x-y)(-csc^2(y)dy/dx)#
Return to the #e^(x-y)# form:
#0= (e^(x-y)-e^(x-y)dy/dx)cot(y)+ e^(x-y)(-csc^2(y)dy/dx)#
Use the distributive property:
#0= e^(x-y)cot(y)-e^(x-y)cot(y)dy/dx- e^(x-y)csc^2(y)dy/dx#
Move the terms containing #dy/dx# to the left:
#e^(x-y)cot(y)dy/dx +e^(x-y)csc^2(y)dy/dx= e^(x-y)cot(y)#
Factor out #dy/dx#:
#(e^(x-y)cot(y) +e^(x-y)csc^2(y))dy/dx= e^(x-y)cot(y)#
#e^(x-y)# is on both sides so it cancels:
#(cot(y) +csc^2(y))dy/dx= cot(y)#
Divide by sides by #cot(y) +csc^2(y)#
#dy/dx= cot(y)/(cot(y) +csc^2(y))#
