x=tan(x+y)
Diff.ing, both sides w.r.t. y, and keeping in mind the Chain Rule,
dx/dy=d/dytan(x+y)=(sec^2(x+y))d/dy(x+y)=sec^2(x+y)*(dx/dy+1)
dx/dy-(sec^2(x+y))dx/dy=sec^2(x+y)
{1-sec^2(x+y)}dx/dy=sec^2(x+y)
Using, sec^2theta=1+tan^2theta, we have,
(-tan^2(x+y))dx/dy=1+tan^2(x+y)
Knowing that, we have tan(x+y)=x.
-x^2dx/dy=1+x^2, giving, dx/dy=-(1+x^2)/x^2
Therefore, dy/dx=-x^2/(1+x^2).
Method II
x=tan(x+y)
rArr d/dxx=d/dx(tan(x+y))
rArr 1=sec^2(x+y){d/dx(x+y)}=sec^2(x+y){1+dy/dx}=(1+x^2){1+dy/dx}
rArr 1+dy/dx=1/(1+x^2) rArr dy/dx=1/(1+x^2)-1=(1-1-x^2)/(1+x^2)
rArr dy/dx=-x^2/(1+x^2), as in Method I !
Method III
x=tan(x+y)
arctanx=x+y rArr arctanx-x=y
rArr dy/dx=1/(1+x^2)-1 =-x^2/(1+x^2), as derived before!
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