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A curve is defined by the parametric equations
x(t)=-7cos(-6t) and
y(t)=9sin(6t). Find
(dy)/(dx).
Answer:

A curve is defined by the parametric equations $x(t)=-7 \cos (-6 t)$ and $y(t)=9 \sin (6 t)$ . Find $\frac{d y}{d x}$ . $\newline$ Answer:

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Write equations of cosine functions using properties

Full solution

Q. A curve is defined by the parametric equations

x(t)=-7 \cos (-6 t)

and

y(t)=9 \sin (6 t)

. Find

\frac{d y}{d x}

.

\newline

Answer:

Find $\frac{dx}{dt}$ : To find $\frac{dy}{dx}$ for parametric equations, we need to find $\frac{dy}{dt}$ and $\frac{dx}{dt}$ separately and then divide $\frac{dy}{dt}$ by $\frac{dx}{dt}$ .
Find $\frac{dy}{dt}$ : First, we find $\frac{dx}{dt}$ . The derivative of $x(t) = -7\cos(-6t)$ with respect to $t$ is $\frac{dx}{dt} = -7 \times 6\sin(-6t)$ because the derivative of $\cos(u)$ with respect to $u$ is $-\sin(u)$ and we apply the chain rule with $u = -6t$ .
$\frac{dx}{dt} = -7 \times 6\sin(-6t) = 42\sin(-6t)$
Calculate $\frac{dy}{dx}$ : Next, we find $\frac{dy}{dt}$ . The derivative of $y(t) = 9\sin(6t)$ with respect to $t$ is $\frac{dy}{dt} = 9 \times 6\cos(6t)$ because the derivative of $\sin(u)$ with respect to $u$ is $\cos(u)$ and we apply the chain rule with $u = 6t$ .
$\frac{dy}{dt} = 9 \times 6\cos(6t) = 54\cos(6t)$
Simplify fraction: Now we divide $\frac{dy}{dt}$ by $\frac{dx}{dt}$ to find $\frac{dy}{dx}$ . $\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{54\cos(6t)}{42\sin(-6t)}$
Final result: We can simplify the fraction by dividing both the numerator and the denominator by $6$ . $\newline$ $\frac{dy}{dx} = \frac{(\frac{54}{6} \cdot \cos(6t))}{(\frac{42}{6} \cdot \sin(-6t))} = \frac{(9\cos(6t))}{(7\sin(-6t))}$
Final result: We can simplify the fraction by dividing both the numerator and the denominator by $6$ . $\newline$ $\frac{dy}{dx} = \frac{(54/6 \cdot \cos(6t))}{(42/6 \cdot \sin(-6t))} = \frac{(9\cos(6t))}{(7\sin(-6t))}$ Since $\sin(-\theta) = -\sin(\theta)$ , we can simplify the denominator to $-\sin(6t)$ . $\newline$ $\frac{dy}{dx} = \frac{(9\cos(6t))}{(7 \cdot -\sin(6t))} = -\frac{9\cos(6t)}{7\sin(6t)}$

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