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

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Answer 1 · 2018-06-21T12:53:24

#dy/dx = y/x#

#y/x = arctan(x/y)#

Differentiate both sides of the equation:

#d/dx (y/x) = d/dx ( arctan(x/y) )#

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#