To find the equation of the curve, we need to integrate the derivative \(\frac{dy}{dx} = 3x^{\frac{1}{2}} - 3x^{-\frac{1}{2}}\).

Integrating term by term:

\(y = \int (3x^{\frac{1}{2}} - 3x^{-\frac{1}{2}}) \, dx\)

\(= \int 3x^{\frac{1}{2}} \, dx - \int 3x^{-\frac{1}{2}} \, dx\)

\(= \frac{3}{\frac{3}{2}} x^{\frac{3}{2}} - \frac{3}{\frac{1}{2}} x^{\frac{1}{2}} + c\)

\(= 2x^{\frac{3}{2}} - 6x^{\frac{1}{2}} + c\)

Using the point (4, 7) to find \(c\):

\(7 = 2(4)^{\frac{3}{2}} - 6(4)^{\frac{1}{2}} + c\)

\(7 = 16 - 12 + c\)

\(7 = 4 + c\)

\(c = 3\)

Thus, the equation of the curve is \(y = 2x^{\frac{3}{2}} - 6x^{\frac{1}{2}} + 3\).