$$\text { Find the value of } d ^ { 2 } y / d x ^ { 2 } \text { at the point defined by the given value of } t \text {. }$$
-$$x = \csc t , y = 2 \cot t , t = \frac { \pi } { 3 }$$
A) $4 \sqrt { 3 }$
B) $- 6 \sqrt { 3 }$
C) $2 \sqrt { 3 }$
D) $- 4 \sqrt { 3 }$

Question

Quizplus · Accepted Answer

To find $d^2y/dx^2$ at $t = \pi/3$, we first find $dy/dx$ by using the chain rule: $dy/dx = (dy/dt) / (dx/dt)$. Given $x = \csc(t)$ and $y = 2\cot(t)$, we find their derivatives with respect to $t$: $dx/dt = -\csc(t)\cot(t)$ and $dy/dt = -2\csc^2(t)$. Thus, $dy/dx = (dy/dt) / (dx/dt) = ( -2\csc^2(t) ) / ( -\csc(t)\cot(t) ) = 2\csc(t)/\cot(t)$.To find $d^2y/dx^2$, we differentiate $dy/dx$ with respect to $t$ and then divide by $dx/dt$: $d^2y/dx^2 = \frac{d}{dt}\left(\frac{dy}{dx}\right) / \frac{dx}{dt}$. Differentiating $2\csc(t)/\cot(t)$ with respect to $t$ involves using the quotient and chain rules, leading to a more complex expression. Substituting $t = \pi/3$ into this expression and simplifying using trigonometric identities yields $-6\sqrt{3}$, which corresponds to choice B.

Find the value of d2y/dx2 at the point defined by the given value of t. \text { Find the value of } d ^ { 2 } y / d x ^ { 2 } \text { at the point defined by the given value of } t \text {. } Find the value of d2y/dx2 at the point defined by the given value of t.

$\text { Find the value of } d ^ { 2 } y / d x ^ { 2 } \text { at the point defined by the given value of } t \text {. }$