To find the gradient of the curve, we need to differentiate the given equation:

\(y = \frac{1}{k}x^{\frac{1}{2}} + x^{-\frac{1}{2}} + \frac{1}{k^2}\).

The derivative \(\frac{dy}{dx}\) is:

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

Substitute \(x = \frac{1}{4}\) into the derivative:

\(\frac{dy}{dx} = \frac{1}{2k}(\frac{1}{4})^{-\frac{1}{2}} - \frac{1}{2}(\frac{1}{4})^{-\frac{3}{2}}\).

Simplify \((\frac{1}{4})^{-\frac{1}{2}} = 2\) and \((\frac{1}{4})^{-\frac{3}{2}} = 8\):

\(\frac{dy}{dx} = \frac{1}{2k} \cdot 2 - \frac{1}{2} \cdot 8\).

\(\frac{dy}{dx} = \frac{1}{k} - 4\).

It is given that \(\frac{dy}{dx} = 3\) when \(x = \frac{1}{4}\):

\(3 = \frac{1}{k} - 4\).

Solving for \(k\):

\(3 + 4 = \frac{1}{k}\).

\(7 = \frac{1}{k}\).

\(k = \frac{1}{7}\).