Consider an ellipse with equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) and a hyperbola with equation \(\frac{x^2}{c^2} - \frac{y^2}{d^2} = 1\). If the derivative of the equation of the ellipse at a point where it intersects the hyperbola is equal to the derivative of the equation of the hyperbola at that same point, and if the ellipse has semi-major axis \(a = 3\) and semi-minor axis \(b = 2\), find the value of \(d\) when \(c = 3\). Let's think step by step and output the final answer within \boxed{}.
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