Which of the following is the derivative (dy)/(dx) for the plane curve defined by the equations x(t)=(1)/(2)sin 2t, y(t)=2cos 2t, and 0 <= t <= 2pi ? Select the correct answer below: (A) 4cot 2t (B) tan 4t (C) -tan 4t (D) -4tan 2t

Find Derivative Process: To find the derivative (dy)/(dx), we need to find dy/dt and dx/dt first and then divide dy/dt by dx/dt.Calculate dx/dt: Calculate dx/dt: dx/dt = d/dt [(1/2)sin(2t)] = (1/2) * d/dt [sin(2t)] = (1/2) * 2cos(2t) = cos(2t)Calculate dy/dt: Calculate dy/dt: dy/dt = d/dt [2cos(2t)] = -2 * d/dt [cos(2t)] = -2 * (-2sin(2t)) = 4sin(2t)Find (dy)/(dx): Now, find (dy)/(dx) by dividing dy/dt by dx/dt: (dy)/(dx) = (dy/dt) / (dx/dt) = (4sin(2t)) / (cos(2t)) = 4tan(2t)Check Answer Choices: However, we need to check the answer choices to see which one matches our result. The correct answer should be in terms of t and should match the form of the derivative we found.Correcting Mistake: The answer choices are: A) 4cot 2t B) tan 4t C) -tan 4t D) -4tan 2t Our result is 4tan(2t), which is not listed. We need to check if we made a mistake or if there is a trigonometric identity that can transform our result into one of the given options.Upon reviewing, we realize that we made a mistake in our calculation. The correct derivative should be in terms of 2t, not 4t. Therefore, the correct answer should be -4tan(2t) because the derivative of y with respect to t is negative, and we missed the negative sign in our previous step.

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Which of the following is the derivative (dy)/(dx) for the plane curve defined by the equations x(t)=(1)/(2)sin 2t, y(t)=2cos 2t, and 0 <= t <= 2pi ?
Select the correct answer below:
(A) 4cot 2t
(B) tan 4t
(C) -tan 4t
(D) -4tan 2t

Which of the following is the derivative $\frac{d y}{d x}$ for the plane curve defined by the equations $x(t)=\frac{1}{2} \sin 2 t$ , $y(t)=2 \cos 2 t$ , and $0 \leq t \leq 2 \pi$ ? $\newline$ Select the correct answer below: $\newline$ (A) $4 \cot 2 t$ $\newline$ (B) $\tan 4 t$ $\newline$ (C) $-\tan 4 t$ $\newline$ (D) $-4 \tan 2 t$

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Q. Which of the following is the derivative

\frac{d y}{d x}

for the plane curve defined by the equations

x(t)=\frac{1}{2} \sin 2 t

y(t)=2 \cos 2 t

, and

0 \leq t \leq 2 \pi

\newline

Select the correct answer below:

\newline

(A)

4 \cot 2 t

\newline

(B)

\tan 4 t

\newline

(C)

-\tan 4 t

\newline

(D)

-4 \tan 2 t

Find Derivative Process: To find the derivative $\frac{dy}{dx}$ , we need to find $\frac{dy}{dt}$ and $\frac{dx}{dt}$ first and then divide $\frac{dy}{dt}$ by $\frac{dx}{dt}$ .
Calculate $\frac{dx}{dt}$ : Calculate $\frac{dx}{dt}$ : $\frac{dx}{dt} = \frac{d}{dt} \left[\frac{1}{2}\sin(2t)\right] = \frac{1}{2} \cdot \frac{d}{dt} [\sin(2t)] = \frac{1}{2} \cdot 2\cos(2t) = \cos(2t)$
Calculate $\frac{dy}{dt}$ : Calculate $\frac{dy}{dt}$ : $\frac{dy}{dt} = \frac{d}{dt} [2\cos(2t)] = -2 \times \frac{d}{dt} [\cos(2t)] = -2 \times (-2\sin(2t)) = 4\sin(2t)$
Find $(\frac{dy}{dx}):</b> Now, find \$(\frac{dy}{dx})$ by dividing $\frac{dy}{dt}$ by $\frac{dx}{dt}:$ $\newline$ (\frac{dy}{dx}) = \frac{(\frac{dy}{dt})}{(\frac{dx}{dt})} = \frac{($4$\sin($2$t))}{(\cos($2$t))} = \(4\tan( $2$ t)
Check Answer Choices: However, we need to check the answer choices to see which one matches our result. The correct answer should be in terms of $t$ and should match the form of the derivative we found.
Correcting Mistake: The answer choices are: $\newline$ A) $4\cot 2t$ $\newline$ B) $\tan 4t$ $\newline$ C) $-\tan 4t$ $\newline$ D) $-4\tan 2t$ $\newline$ Our result is $4\tan(2t)$ , which is not listed. We need to check if we made a mistake or if there is a trigonometric identity that can transform our result into one of the given options.
Correcting Mistake: The answer choices are: $\newline$ A) $4\cot 2t$ $\newline$ B) $\tan 4t$ $\newline$ C) $-\tan 4t$ $\newline$ D) $-4\tan 2t$ $\newline$ Our result is $4\tan(2t)$ , which is not listed. We need to check if we made a mistake or if there is a trigonometric identity that can transform our result into one of the given options.Upon reviewing, we realize that we made a mistake in our calculation. The correct derivative should be in terms of $2t$ , not $4t$ . Therefore, the correct answer should be $-4\tan(2t)$ because the derivative of $y$ with respect to $t$ is negative, and we missed the negative sign in our previous step.

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