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A curve is defined by the parametric equations
x(t)=t^(2)+9t and
y(t)=-3t^(2)-10 t+4. Find
(dy)/(dx).
Answer:

A curve is defined by the parametric equations $x(t)=t^{2}+9 t$ and $y(t)=-3 t^{2}-10 t+4$ . Find $\frac{d y}{d x}$ . $\newline$ Answer:

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Simplify expressions using trigonometric identities

Full solution

Q. A curve is defined by the parametric equations

x(t)=t^{2}+9 t

and

y(t)=-3 t^{2}-10 t+4

. Find

\frac{d y}{d x}

.

\newline

Answer:

Find $\frac{dx}{dt}$ : To find $\frac{dy}{dx}$ for parametric equations, we first need to find $\frac{dx}{dt}$ and $\frac{dy}{dt}$ separately.
Find $\frac{dy}{dt}$ : The derivative of $x(t)$ with respect to $t$ is $\frac{dx}{dt}$ . We calculate this by differentiating $x(t)=t^2+9t$ with respect to $t$ .
$\frac{dx}{dt} = \frac{d(t^2)}{dt} + \frac{d(9t)}{dt} = 2t + 9$ .
Calculate $\frac{dy}{dx}$ : Similarly, the derivative of $y(t)$ with respect to $t$ is $\frac{dy}{dt}$ . We calculate this by differentiating $y(t)=-3t^2-10t+4$ with respect to $t$ .
$\frac{dy}{dt} = \frac{d(-3t^2)}{dt} + \frac{d(-10t)}{dt} + \frac{d(4)}{dt} = -6t - 10$ .
Simplify $\frac{dy}{dx}$ : Now, we find $\frac{dy}{dx}$ by dividing $\frac{dy}{dt}$ by $\frac{dx}{dt}$ . $\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{-6t - 10}{2t + 9}.$
Simplify $\frac{dy}{dx}$ : Now, we find $\frac{dy}{dx}$ by dividing $\frac{dy}{dt}$ by $\frac{dx}{dt}$ . $\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{-6t - 10}{2t + 9}$ .We simplify the expression for $\frac{dy}{dx}$ if possible. In this case, the expression is already simplified.

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