Let x and y be functions of t with y = x^(1/3). If dx/dt = 7, what is dy/dt when x = 8? Write an exact, simplified answer. ______

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

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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.

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Algebra 2

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 $\frac{dy}{dt}$ .
$\frac{dy}{dt} = \left(\frac{1}{3}\right) * 0.25 * 7 = 0.583333\ldots$