$$\text { Use implicit differentiation to find } d y / d x \text { and } \mathrm { d } ^ { 2 } y / d x ^ { 2 } \text {. }$$
-$$x ^ { 3 / 5 } + y ^ { 3 / 5 } = 8$$
A) $\frac { \mathrm { dy } } { \mathrm { dx } } = \frac { \mathrm { x } ^ { 2 / 5 } } { \mathrm { y } ^ { 2 / 5 } } ; \frac { \mathrm { d } ^ { 2 } \mathrm { y } } { \mathrm { dx } } = - \frac { 2 \mathrm { x } ^ { 3 / 5 } + 2 \mathrm { y } ^ { 3 / 5 } } { 5 \mathrm { x } ^ { 1 / 5 } \mathrm { y } ^ { 7 / 5 } }$
B) $\frac { d y } { d x } = \frac { y ^ { 2 / 5 } } { x ^ { 2 / 5 } } ; \frac { d ^ { 2 } y } { d x ^ { 2 } } = - \frac { 2 x ^ { 3 / 5 } + 2 y ^ { 3 / 5 } } { 5 x ^ { 7 / 5 } y ^ { 1 / 5 } }$
C) $\frac { d y } { d x } = - \frac { y ^ { 2 / 5 } } { x ^ { 2 / 5 } } ; \frac { d ^ { 2 } y } { d x ^ { 2 } } = \frac { 2 y ^ { 1 / 5 } - 2 x } { 5 x ^ { 7 / 5 } y ^ { 3 / 5 } }$
D) $\frac { d y } { d x } = - \frac { y ^ { 2 / 5 } } { x ^ { 2 / 5 } } ; \frac { d ^ { 2 } y } { d x ^ { 2 } } = \frac { 2 x ^ { 3 / 5 } + 2 y ^ { 3 / 5 } } { 5 x ^ { 7 / 5 } y ^ { 1 / 5 } }$

Question

Quizplus · Accepted Answer

Differentiating the given equation implicitly with respect to $x$, we get $\frac{3}{5}x^{-2/5} + \frac{3}{5}y^{-2/5}\frac{dy}{dx} = 0$, which simplifies to $\frac{dy}{dx} = -\frac{y^{2/5}}{x^{2/5}}$. For the second derivative, further differentiation and simplification yield $\frac{d^2y}{dx^2} = \frac{2x^{3/5} + 2y^{3/5}}{5x^{7/5}y^{1/5}}$.

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.

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