If y/x= arctan(x/y), then dy/dx= ?

1 Answer
Jun 21, 2018

dy/dx = y/xdydx=yx

Explanation:

y/x = arctan(x/y)yx=arctan(xy)

Differentiate both sides of the equation:

d/dx (y/x) = d/dx ( arctan(x/y) )ddx(yx)=ddx(arctan(xy))

use the quotient rule at the first member and the chain rule at the second member:

(xy'-y)/x^2 = 1/(1+x^2/y^2) d/dx (x/y)

(xy'-y)/x^2 = 1/(1+x^2/y^2) (y-xy')/y^2

(xy'-y)/x^2 = (y-xy')/(x^2+y^2)

y'(1/x+x/(x^2+y^2)) = y (1/x^2+1/(x^2+y^2))

y'(1/x+x/(x^2+y^2)) = y/x (1/x+x/(x^2+y^2))

dy/dx = y/x