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In calculus, understanding derivatives is fundamental to analyzing the behavior of functions. The derivative of a function, represented as \( \frac{dy}{dt} \), describes the rate at which the function's value changes with respect to a change in the input variable. In simple terms, it's like measuring how fast something is moving or changing. For the sigmoid function \( y = \frac{50}{1 + 6e^{-2t}} \), calculating the derivative helps us understand how steep the curve is at any given point \( t \). The steeper the curve, the greater the derivative. This is crucial for finding points of maximum steepness or incline on the graph of the function, which is precisely what our problem requires.

A sigmoid function is a type of mathematical function that produces an S-shaped curve. It's commonly used in fields like machine learning and neural networks because of its smooth and bounded nature. The specific sigmoid function we are looking at is \( y = \frac{50}{1 + 6e^{-2t}} \), which is defined for \( t \geq 0 \).
These functions are so named because their graphs resemble a stretched "S" shape. This unique shape occurs because, as \( t \) approaches infinity, the function approaches a maximum value, and as \( t \) approaches negative infinity, the function approaches a minimum value. For our purpose, the interest is in determining where the function's slope is the greatest, which corresponds to its steepest point.

Critical points are important in calculus because they offer insights about the function's extremities, such as maximum, minimum, or saddle points. To find a critical point, you typically look where the first derivative equals zero or is undefined. In our task, however, we seek where the rate of change of the derivative itself is maximized, which involves setting the second derivative to zero.
For the sigmoid function in question, \( y = \frac{50}{1 + 6e^{-2t}} \), we determine the critical point by solving \( \frac{d^2y}{dt^2} = 0 \). This step ensures we're identifying the steepest part of the curve. The result, \( t = \ln(6)/2 \), reflects where this change reaches its peak, and thus, where the graph is steepest.

The second derivative of a function, denoted \( \frac{d^2y}{dt^2} \), provides information about the concavity of the function and the behavior of its first derivative. It tells us how the rate of change of the function is itself changing. This can indicate whether a function is curving up or down, much like how acceleration measures how velocity changes over time.
In the search for the steepest point of our sigmoid curve, we calculate the second derivative to identify where the first derivative itself is changing most rapidly. By setting \( \frac{d^2y}{dt^2} = 0 \), and resolving this equation, we pinpoint the exact \( t \)-value where the steepness is maximized. This step is crucial for accurate analysis and helps in discovering the peak rate of increase, leading us to our exact coordinates of interest on the graph.