if x2 + xy + y3 = 1 find the value of y”’  at the point where x = 1

you are watching: If x2 + xy + y3 = 1 find the value of y”’ at the point where x = 1.

consider the equation x +xy+y-1 need to find by using implicit differentiation. differentiate with respect to x, get dv dh dr dh treat y as a
work of x and use chain rule, get dy 2x(1)+x+y(1)+3y dr dr since _ ( constant )-0 and (x” ) = nxn-1 (2x + y)-(x+3y*) =0 dr d 2rty) -(2x+v -(2x+y) x+3y

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2x y again differentiate v- with respect to x by treating y as constant. (thy x +3 du d d-(2x+y) ахах since (x+3y2) [-(2x + y)]-[-(2x+y)] (x+3y2) x +3 r +3 ( d (x+3y) r +3 t + ах 2x+ y x +3y dx (x+3y’ (x+3y

-(2x+ y) x+3y x+3y again differentiate v” with respect to x by treating x+3y y as constant. (x+3) x+3 dr 18y4-24x2y-24m2 + 2×2 + 2xy-(x + 3y2 dhx

s dy_24x2 – 48y-24y -48xy dy x +3y dhx dhr dhx x+3y r +3 72y3-24×2 – 48xy+ 2r x +3 (x2-46y-24+4 x+3y 72y3-24×2 – 48xy+ 2r x+3y -(2r+y) x +3

find the value of y when x-+19176223089(x)-У3-1 so the only real number which satisfies this equation is y-0. now substitute x,v 0 in the equation v” and simplify -(2x+y) x +3 72y -24×2 -48xy+2x – 46 y 24y’ +4x -(2x+ y x+3y x+3y

(1+3(0′!(-+19176223089)(噐 +19176223089-+19176223089) +19176223089)+19176223089) +19176223089) [(-+19176223089)]-미(+19176223089) [h1 = 42

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