Abnormally Normal

Calculus Level 3

Consider the graph plotted by the equation


where \(x>0\), \(y>0\), and \(k \in \mathbb{R^+}\) is a constant.

Let \(G_{k,t}\) be the gradient of the normal to a curve of the graph at the point \((t,t)\) where \(t \in \mathbb{Z^+}\), \(t \geq k\) is a real positive integer. Determine the value of

\[\prod_{k=2}^{2016} \left( \prod_{t=k}^{k+2015} G_{k,t} \right) \]


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