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curve is defined by the parametric equations
x(t)=7e^(-8t) and
y(t)=-7sin(-10 t). Find
(dy)/(dx).

A curve is defined by the parametric equations $x(t) = 7e^{-8t}$ and $y(t) = -7\sin(-10t)$ . Find $\frac{dy}{dx}$ .

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Find derivatives of using multiple formulae

Full solution

Q. A curve is defined by the parametric equations

x(t) = 7e^{-8t}

and

y(t) = -7\sin(-10t)

. Find

\frac{dy}{dx}

.

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, let's find $\frac{dx}{dt}$ , which is the derivative of $x$ with respect to $t$ .
$\frac{dx}{dt} = \frac{d}{dt} [7e^{(-8t)}]$
Using the chain rule, we get:
$\frac{dx}{dt} = 7 \cdot (-8) \cdot e^{(-8t)}$
$\frac{dx}{dt} = -56e^{(-8t)}$
Calculate $\frac{dy}{dx}$ : Next, we find $\frac{dy}{dt}$ , which is the derivative of $y$ with respect to $t$ .
$\frac{dy}{dt} = \frac{d}{dt} [-7\sin(-10t)]$
Using the chain rule, we get:
$\frac{dy}{dt} = -7 \times (-10) \times \cos(-10t)$
$\frac{dy}{dt} = 70\cos(-10t)$
Since cosine is an even function, $\cos(-10t) = \cos(10t)$ , so we can simplify to:
$\frac{dy}{dt} = 70\cos(10t)$
Simplify expression: Now we have both derivatives: $\newline$ $\frac{dx}{dt} = -56e^{-8t}$ $\newline$ $\frac{dy}{dt} = 70\cos(10t)$ $\newline$ To find $\frac{dy}{dx}$ , we divide $\frac{dy}{dt}$ by $\frac{dx}{dt}$ : $\newline$ $\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}$ $\newline$ $\frac{dy}{dx} = \frac{70\cos(10t)}{-56e^{-8t}}$
Final $(\frac{dy}{dx})$ expression: We can simplify the expression by dividing both the numerator and the denominator by $7$ : $(\frac{dy}{dx}) = \frac{10\cos(10t)}{-8e^{-8t}}$
Final $(\frac{dy}{dx})$ expression: We can simplify the expression by dividing both the numerator and the denominator by $7$ :
$(\frac{dy}{dx}) = \frac{10\cos(10t)}{-8e^{-8t}}$ Finally, we can write the simplified expression for $(\frac{dy}{dx})$ :
$(\frac{dy}{dx}) = -\frac{10\cos(10t)}{8e^{-8t}}$

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