Question

If tan-1 (x+y)/(1-xy)=k show that dy/dx=-(1+y2)/(1+x2)

Answer 1

Step-by-step explanation:

We have,

$\tan ^{ - 1} \bigg( \frac{x + y}{1 - xy} \bigg) = k$

$\implies \tan ^{ - 1} (x) + \tan ^{ - 1} (y) = k$

Differentiating both sides w.r.t x,

$\implies \frac{1}{1 + {x}^{2} } + \frac{1}{1 + {y}^{2} } \frac{dy}{dx} = 0$

$\implies \frac{1}{1 + {y}^{2} } \frac{dy}{dx} = - \frac{1}{1 + {x}^{2} }$

$\implies \frac{dy}{dx} = - \frac{1 + {y}^{2} }{1 + {x}^{2} }$