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Let $x$ and $y$ be functions of $t$ with $y = 4\sin x$ . If $\frac{dx}{dt} = 5$ , what is $\frac{dy}{dt}$ when $x = \frac{\pi}{3}$ ? $\newline$ Write an exact, simplified answer.

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Write and solve direct variation equations

Full solution

Q. Let

x

and

y

be functions of

t

with

y = 4\sin x

. If

\frac{dx}{dt} = 5

, what is

\frac{dy}{dt}

when

x = \frac{\pi}{3}

?

\newline

Write an exact, simplified answer.

Identify Function and Differentiate: Identify the function and differentiate it with respect to $x$ . $\newline$ Given $y = 4\sin(x)$ , differentiate to find $\frac{dy}{dx}$ . $\newline$ $\frac{dy}{dx} = 4\cos(x)$
Use Chain Rule for $\frac{dy}{dt}$ : Use the chain rule to find $\frac{dy}{dt}$ . Given $\frac{dx}{dt} = 5$ , use the chain rule: $\frac{dy}{dt} = \frac{dy}{dx} \cdot \frac{dx}{dt}$ . $\frac{dy}{dt} = 4\cos(x) \cdot 5$
Substitute $x$ into $\frac{dy}{dt}$ : Substitute $x = \frac{\pi}{3}$ into the equation. $\newline$ Substitute $x = \frac{\pi}{3}$ into $\frac{dy}{dt} = 4\cos(x) \times 5$ . $\newline$ $\frac{dy}{dt} = 4\cos\left(\frac{\pi}{3}\right) \times 5$ $\newline$ $\frac{dy}{dt} = 4 \times \left(\frac{1}{2}\right) \times 5$ $\newline$ $\frac{dy}{dt} = 10$

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