$$\text { Use implicit differentiation to find } d y / d x \text { and } \mathrm { d } ^ { 2 } y / d x ^ { 2 } \text {. }$$
-$x y + 3 = y$, at the point $( 4 , - 1 )$
A) $\frac { \mathrm { dy } } { \mathrm { dx } } = \frac { 1 } { 3 } ; \frac { \mathrm { d } ^ { 2 } \mathrm { y } } { \mathrm { dx } { } ^ { 2 } } = \frac { 2 } { 9 }$
B) $\frac { d y } { d x } = 3 ; \frac { d ^ { 2 } y } { d x ^ { 2 } } = - 24$
C) $\frac { \mathrm { dy } } { \mathrm { dx } } = \frac { 1 } { 3 } ; \frac { \mathrm { d } ^ { 2 } \mathrm { y } } { \mathrm { dx } ^ { 2 } } = - \frac { 2 } { 9 }$
D) $\frac { d y } { d x } = - \frac { 1 } { 3 } ; \frac { d ^ { 2 } y } { d x ^ { 2 } } = 0$

Question

Quizplus · Accepted Answer

We start by taking the derivative of both sides with respect to x, using the product rule on the left-hand side:
\begin{align*}
\frac{d}{dx}(xy+3)&=\frac{d}{dx}y\
y+x\frac{dy}{dx}+0&= \frac{dy}{dx}\
\frac{dy}{dx}(1-x)&=-y\
\frac{dy}{dx}&=-\frac{y}{1-x}
\end{align*}
At the point (4,-1), we have $x=4$ and $y=-1$, so 
$$\frac{dy}{dx}=-\frac{-1}{1-4}=\frac{1}{3}$$
To find the second derivative, we need to use the quotient rule:
\begin{align*}
\frac{d}{dx}\left(-\frac{y}{1-x}ight)&=\frac{(-\frac{dy}{dx}(1-x)-(-y)(-1))}{(1-x)^2}\
&=-\frac{(1/3)(1-4)-(-1)(-1)}{(1-4)^2}\
&=-\frac{2}{9}
\end{align*}
Therefore, the answer is (C).

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.

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