Given the curve \(y = 2x + \frac{5}{x}\), we need to find \(\frac{dy}{dt}\) when \(x = 1\) and \(\frac{dx}{dt} = 0.02\).

First, differentiate \(y\) with respect to \(x\):

\(\frac{dy}{dx} = \frac{d}{dx}(2x + \frac{5}{x}) = 2 - \frac{5}{x^2}\).

Substitute \(x = 1\) into \(\frac{dy}{dx}\):

\(\frac{dy}{dx} = 2 - \frac{5}{1^2} = 2 - 5 = -3\).

Now, use the chain rule to find \(\frac{dy}{dt}\):

\(\frac{dy}{dt} = \frac{dy}{dx} \cdot \frac{dx}{dt}\).

Substitute \(\frac{dy}{dx} = -3\) and \(\frac{dx}{dt} = 0.02\):

\(\frac{dy}{dt} = -3 \times 0.02 = -0.06\).