To find \(f'(x)\), we use the quotient rule for differentiation. The function \(f(x) = \frac{15}{2x+3}\) can be rewritten as \(f(x) = 15(2x+3)^{-1}\).

The derivative of \(f(x)\) is:

\(f'(x) = -15(2x+3)^{-2} \cdot 2 = \frac{-30}{(2x+3)^2}\).

Since \((2x+3)^2\) is always positive, \(f'(x) < 0\) for all \(x\) in the domain \(0 \leq x \leq 6\). This means \(f(x)\) is strictly decreasing, and therefore, \(f\) has an inverse.