$$\text { Use implicit differentiation to find } d y / d x \text { and } \mathrm { d } ^ { 2 } y / d x ^ { 2 } \text {. }$$
-$$y ^ { 2 } - x ^ { 2 } = 4$$
A) $\frac { \mathrm { dy } } { \mathrm { dx } } = - \frac { \mathrm { x } } { \mathrm { y } } ; \frac { \mathrm { d } ^ { 2 } \mathrm { y } } { \mathrm { dx } ^ { 2 } } = \frac { \mathrm { y } ^ { 2 } - \mathrm { x } ^ { 2 } } { \mathrm { y } ^ { 3 } }$
B) $\frac { \mathrm { dy } } { \mathrm { dx } } = \frac { \mathrm { x } } { \mathrm { y } } ; \frac { \mathrm { d } ^ { 2 } \mathrm { y } } { \mathrm { dx } 2 } = \frac { \mathrm { y } ^ { 2 } - \mathrm { x } ^ { 2 } } { \mathrm { y } ^ { 2 } }$
C) $\frac { d y } { d x } = \frac { x } { y } ; \frac { d ^ { 2 } y } { d x ^ { 2 } } = \frac { y ^ { 2 } - x ^ { 2 } } { y ^ { 3 } }$
D) $\frac { \mathrm { dy } } { \mathrm { dx } } = \frac { \mathrm { x } } { \mathrm { y } } ; \frac { \mathrm { d } ^ { 2 } \mathrm { y } } { \mathrm { dx } \mathrm { x } ^ { 2 } } = \frac { \mathrm { y } - \mathrm { x } ^ { 2 } } { \mathrm { y } ^ { 2 } }$

Question

Quizplus · Accepted Answer

We have the equation: $y^2 - x^2 = 4$

Taking the derivative with respect to $x$, we get:

$2y\cdot \frac{dy}{dx} - 2x = 0$

$\frac{dy}{dx} = \frac{x}{y}$

Taking the derivative of $\frac{dy}{dx}$ with respect to $x$, we get:

$\frac{d}{dx}\left(\frac{x}{y}\right) = \frac{y\cdot 1 - x\cdot\frac{dy}{dx}}{y^2}$

$\frac{d^2y}{dx^2} = \frac{y^2 - x^2}{y^3}$

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.

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