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Let $x$ and $y$ be functions of $t$ with $y = 4e^x$ . If $\frac{dx}{dt} = \frac{1}{4}$ , what is $\frac{dy}{dt}$ when $x = 2$ ? $\newline$ 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 = 4e^x

. If

\frac{dx}{dt} = \frac{1}{4}

, what is

\frac{dy}{dt}

when

x = 2

?

\newline

Write an exact, simplified answer.

Identify Relationship: Identify the relationship and differentiate. $\newline$ Given $y = 4e^x$ and $\frac{dx}{dt} = \frac{1}{4}$ , use the chain rule to find $\frac{dy}{dt}$ . $\newline$ $\frac{dy}{dt} = \frac{dy}{dx} \cdot \frac{dx}{dt} = 4e^x \cdot \left(\frac{1}{4}\right)$
Use Chain Rule: Substitute $x = 2$ into the differentiated equation. $\newline$ $\frac{dy}{dt} = 4e^{2} \cdot \left(\frac{1}{4}\right) = e^{2}$

More problems from Write and solve direct variation equations

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In a direct variation,

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Question

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\_\_\_\_

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Question

z

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and inversely with

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y = -12

x = -4

z = -36

. What is the equation of variation? Write your answer in the form

A

\frac{A}{B}

A

B

are constants or variable expressions. Remember to include the value of

k

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\newline

\ z

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Question

Which equation describes this relationship? Remember to include

k

, the constant of variation.

a

varies directly with

b

and inversely with

c

d

\newline

\newline

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

\newline

\[[B]a = \frac{kbd}{c}\]

\newline

\[[C]a = \frac{k}{bcd}\]

\newline

\[[D]a = \frac{kd}{bc}\]

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Question

z

varies directly with

y

and inversely with

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x

y = 12

w = 8

x = –5

z = –3

. What is the value of

k

, the constant of variation?

\newline

Write your answer in the form

A

\frac {(A)}{(B)}

A

B

are constants or variable expressions.

\newline

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Question

Given the substitutions

\ln 2=a, \ln 3=b

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\newline

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Question

Square VWXY is dilated by a scale factor of

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V^{\prime} W^{\prime} X^{\prime} Y^{\prime}

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Question

b

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\begin{array}{l}\frac{10}{8}=\frac{b}{12} \\b=\square\end{array}

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Question

In a direct variation,

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Write your answer as an equation with

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Question

In a direct variation,

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Write your answer as an equation with

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