$$\text { Use implicit differentiation to find } d y / d x \text { and } \mathrm { d } ^ { 2 } y / d x ^ { 2 } \text {. }$$
-$$2 y - x + x y = 7$$
A) $\frac { \mathrm { dy } } { \mathrm { dx } } = \frac { 1 - \mathrm { y } } { 2 + \mathrm { x } } ; \frac { \mathrm { d } ^ { 2 } \mathrm { y } } { \mathrm { dx } ^ { 2 } } = \frac { 2 \mathrm { y } - 2 } { ( 2 + \mathrm { x } ) ^ { 2 } }$
B) $\frac { d y } { d x } = \frac { y + 1 } { x + 2 } ; \frac { d ^ { 2 } y } { d x ^ { 2 } } = \frac { 2 y + 2 } { ( x + 2 ) ^ { 2 } }$
C) $\frac { \mathrm { dy } } { \mathrm { dx } } = - \frac { 1 + \mathrm { y } } { \mathrm { x } + 2 } ; \frac { \mathrm { d } ^ { 2 } \mathrm { y } } { \mathrm { dx } ^ { 2 } } = \frac { \mathrm { y } + 1 } { ( 2 + \mathrm { x } ) ^ { 2 } }$
D) $\frac { \mathrm { dy } } { \mathrm { dx } } = - \frac { 1 + \mathrm { y } } { \mathrm { x } + 2 } ; \frac { \mathrm { d } ^ { 2 } \mathrm { y } } { \mathrm { dx } ^ { 2 } } = \frac { 2 \mathrm { y } - 2 } { ( \mathrm { x } + 2 ) ^ { 2 } }$

Question

Quizplus · Accepted Answer

Implicit differentiation gives:$$\frac{d}{dx}[2y-x+xy]=\frac{d}{dx}[7]$$$$2\frac{dy}{dx}-1+y+x\frac{dy}{dx}=0$$$$\frac{dy}{dx}=\frac{1-y}{2+x}$$Differentiating again:$$\frac{d^2y}{dx^2}=\frac{-(2+x)\frac{dy}{dx}-(1-y)(1)}{(2+x)^2}$$Substituting for $\frac{dy}{dx}$:$$\frac{d^2y}{dx^2}=\frac{-(2+x)\left(\frac{1-y}{2+x}\right)-(1-y)(1)}{(2+x)^2}$$$$\frac{d^2y}{dx^2}=\frac{2y-2}{(2+x)^2}$$

Use implicit differentiation to find dy/dx and d2y/dx2. \text { Use implicit differentiation to find } d y / d x \text { and } \mathrm { d } ^ { 2 } y / d x ^ { 2 } \text {. } Use implicit differentiation to find dy/dx and d2y/dx2.

$\text { Use implicit differentiation to find } d y / d x \text { and } \mathrm { d } ^ { 2 } y / d x ^ { 2 } \text {. }$