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

. If

\frac{dx}{dt} = 4

, what is

\frac{dy}{dt}

when

x = 1

?

\newline

Write an exact, simplified answer.

Identify Relationship: Identify the relationship between $y$ and $x$ . $\newline$ Given $y = x^{(1/2)}$ , which implies $y$ is the square root of $x$ .
Use Chain Rule: Use the chain rule to find $\frac{dy}{dt}$ . $\frac{dy}{dt} = \left(\frac{dy}{dx}\right) \cdot \left(\frac{dx}{dt}\right)$ . Since $\frac{dy}{dx}$ is the derivative of $x^{\frac{1}{2}}$ , which is $\left(\frac{1}{2}\right)x^{-\frac{1}{2}}$ , and $\frac{dx}{dt} = 4$ , substitute these values.
Calculate $\frac{dy}{dx}$ : Calculate $\frac{dy}{dx}$ when $x = 1$ . $\frac{dy}{dx} = \left(\frac{1}{2}\right)\left(1^{-\frac{1}{2}}\right) = \left(\frac{1}{2}\right)(1) = \frac{1}{2}$ .
Substitute Values: Substitute values to find $\frac{dy}{dt}$ . $\newline$ $\frac{dy}{dt} = \left(\frac{1}{2}\right) * 4 = 2$ .

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