Implicit Differentiation

romsek

Math Team
Sep 2015
2,875
1,608
USA
$\sqrt{x+y} + xy = 21\\
\dfrac{\partial}{\partial x} \left(\sqrt{x+y} + xy\right) =\dfrac{\partial}{\partial x} 21 = 0\\
\dfrac{1+y'}{2\sqrt{x+y}}+ y+xy' = 0\\

\left(\dfrac{1}{2\sqrt{x+y}}+y\right) + \left(\dfrac{1}{2\sqrt{x+y}}+x\right)y'=0\\

y' = -\dfrac{\dfrac{1}{2\sqrt{x+y}}+y}{\dfrac{1}{2\sqrt{x+y}}+x}\\~\\

y' = -\dfrac{1+2y\sqrt{x+y}}{1+2x\sqrt{x+y}}

$
 
Jun 2015
2
0
Philippines
$\sqrt{x+y} + xy = 21\\
\dfrac{\partial}{\partial x} \left(\sqrt{x+y} + xy\right) =\dfrac{\partial}{\partial x} 21 = 0\\
\dfrac{1+y'}{2\sqrt{x+y}}+ y+xy' = 0\\

\left(\dfrac{1}{2\sqrt{x+y}}+y\right) + \left(\dfrac{1}{2\sqrt{x+y}}+x\right)y'=0\\

y' = -\dfrac{\dfrac{1}{2\sqrt{x+y}}+y}{\dfrac{1}{2\sqrt{x+y}}+x}\\~\\

y' = -\dfrac{1+2y\sqrt{x+y}}{1+2x\sqrt{x+y}}

$
Thank you so much for your kindness ma'am/sir.