Question

given ( y = sqrt{x} ). find ( \frac{dx}{dt} ) when ( y = 4 ) and ( \frac{dy}{dt} = 1.95 ).

( \frac{dx}{dt} = square ) (simplify your answer.)

Explanation:

Step1: Express \(x\) in terms of \(y\)

Given \(y = \sqrt{x}\), we can rewrite it as \(x=y^{2}\).

Step2: Differentiate \(x\) with respect to \(t\)

Using the chain - rule \(\frac{dx}{dt}=2y\frac{dy}{dt}\).

Step3: Substitute the given values of \(y\) and \(\frac{dy}{dt}\)

When \(y = 4\) and \(\frac{dy}{dt}=1.95\), we substitute these values into the equation \(\frac{dx}{dt}=2y\frac{dy}{dt}\).
So \(\frac{dx}{dt}=2\times4\times1.95\).
First, calculate \(2\times4 = 8\), then \(8\times1.95=15.6\).

Answer:

\(15.6\)