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Let $x$ and $y$ be functions of $t$ with $y = 7x$ . If $\frac{dx}{dt} = 2$ , what is $\frac{dy}{dt}$ when $x = \frac{\pi}{2}$ ? $\newline$ 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 = 7x

. If

\frac{dx}{dt} = 2

, what is

\frac{dy}{dt}

when

x = \frac{\pi}{2}

?

\newline

Write an exact, simplified answer.

Identify Relationship: Identify the relationship between $y$ and $x$ . Given $y = 7x$ , differentiate both sides with respect to $t$ to find $\frac{dy}{dt}$ .
Differentiate with Respect: Differentiate $y = 7x$ with respect to $t$ . Using the chain rule, $\frac{dy}{dt} = 7 \cdot \frac{dx}{dt}$ .
Substitute Value of $dx/dt$ : Substitute the value of $dx/dt$ . Given $dx/dt = 2$ , substitute into $dy/dt = 7 \times dx/dt$ to get $dy/dt = 7 \times 2 = 14$ .
Check Value of x: Check the value of $x$ . Since $x = \frac{\pi}{2}$ does not affect the differentiation directly and we are given $\frac{dx}{dt}$ , no need to use this value in our differentiation.

More problems from Write and solve direct variation equations

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Question

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\[[A]a = \frac{kb}{cd}\]

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In the equation shown,

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3

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