dy/dx=-x^2/ (1+x^2). 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).

dy dx = sec2x Explanation: y = tan(x) tanx = sinx cosx y + Δy = tan(x + Δx) tan(x + Δx) = sin(x +Δx) cos(x + Δx)