Resources

Resources

Bytelearn - cat image with glasses

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Let $x$ and $y$ be functions of $t$ with $y = x^{\frac{1}{3}}$ . If $\frac{dx}{dt} = 7$ , what is $\frac{dy}{dt}$ when $x = 8$ ? $\newline$ Write an exact, simplified answer.

View step-by-step help

View step-by-step help

Write and solve direct variation equations

Full solution

Q. Let

x

and

y

be functions of

t

with

y = x^{\frac{1}{3}}

. If

\frac{dx}{dt} = 7

, what is

\frac{dy}{dt}

when

x = 8

?

\newline

Write an exact, simplified answer.

Identify Relationship: Identify the relationship and differentiate using the chain rule. $\newline$ Given $y = x^{\frac{1}{3}}$ , differentiate both sides with respect to $t$ . $\newline$ $\frac{dy}{dt} = \frac{1}{3}x^{-\frac{2}{3}} \cdot \frac{dx}{dt}$
Differentiate Using Chain Rule: Substitute the given values into the differentiated equation. $\newline$ $\frac{dx}{dt} = 7$ and $x = 8$ . $\newline$ $\frac{dy}{dt} = \left(\frac{1}{3}\right)(8)^{-\frac{2}{3}} \times 7$
Substitute Given Values: Calculate the value of $x^{(-2/3)}$ when $x = 8$ . $\newline$ $8^{(-2/3)} = 1/(8^{(2/3)}) = 1/(4) = 0.25$
Calculate $x^{-\frac{2}{3}}$ : Finish the calculation for $\frac{dy}{dt}$ .
$\frac{dy}{dt} = \left(\frac{1}{3}\right) * 0.25 * 7 = 0.583333\ldots$

More problems from Write and solve direct variation equations

Question

In a direct variation,

y=15

x=5

. Write a direct variation equation that shows the relationship between

x

y

. Write your answer as an equation with

y

first, followed by an equals sign.

Posted 2 months ago

Question

In an inverse variation,

y = 4

x = 6

. Write an inverse variation equation that shows the relationship between

x

y

. Write the equation using a decimal or an integer.

\_\_\_\_

Posted 2 months ago

Question

z

varies directly with

y

and inversely with

x

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

, the constant of variation, in exact form.

\newline

\ z

Posted 2 months ago

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

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

Posted 2 months ago

Question

z

varies directly with

y

and inversely with

w

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

Posted 2 months ago

Question

Given the substitutions

\ln 2=a, \ln 3=b

\ln 5=c

, find the value of

\ln \left(\frac{e}{3}\right)

a, b

c

\newline

Posted 2 months ago

Question

Square VWXY is dilated by a scale factor of

6

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

. Side W'X' measures

12

. What is the measure of side WX?

\newline

Posted 2 months ago

Question

6

2

\newline

2

4

2

11

1

2

16

2

\newline

In the system of equations, r is a constant. For what value of r does the system of linear equations have infinitely many solutions?

Posted 2 months ago

Question

1

5

\newline

Use the given information to find

m \angle P Q W

\newline

2

\newline

m \angle P Q W

m \angle W Q R=134 x

m \angle P Q R=169^{\circ}

m \angle P Q W=36 x-1

Posted 2 months ago

Question

10+7 x=4 x+2+b x

\newline

In the equation shown,

b

is a constant. For what value of

b

does the equation have no solutions?

\newline

1

\newline

3

\newline

5

\newline

6

\newline

7

Posted 2 months ago