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Differentiation is a fundamental concept in calculus. It involves finding the derivative of a function, which represents the rate of change of the function with respect to a variable. In simpler terms, the derivative tells us how the function's value changes as its input changes.
For example, in the given exercise, we start with the function: \[ f(t) = t^2 - \frac{1}{2} \]To find its first derivative, we apply the differentiation rules. We differentiate each term separately:\[ \frac{d}{dt} (t^2 - \frac{1}{2}) = \frac{d}{dt}(t^2) - \frac{d}{dt}(\frac{1}{2}) \]Using the power rule, the derivative of \[t^2\] is \[2t\], and the derivative of a constant \[- \frac{1}{2}\] is 0. So, the first derivative is:\[ f'(t) = 2t\]Differentiation is a vital skill in solving various mathematical problems, including those involving rates of change and slopes of curves.

Verification of solutions is essential when working with differential equations. It means checking if a given function satisfies the differential equation. By substituting the function and its derivatives back into the original equation, we determine if both sides of the equation are equal.
In our exercise, after finding the first derivative of the function, we substitute it into the differential equation to verify if it holds true. The differential equation given is:\[\big(y'\big)^2 - 4y = 2\]Let's substitute \[y = t^2 - \frac{1}{2}\] and its derivative \[ y' = 2t \]:\[\big(2t\big)^2 - 4\big(t^2 - \frac{1}{2}\big)\]By simplifying this expression, we confirm that:\[4t^2 - 4t^2 + 2 = 2\]Since both sides of the equation are equal, we've verified that \[ f(t) = t^2 - \frac{1}{2} \] is indeed a solution. This process ensures that our solution is correct and aligns with the differential equation's requirements.

The first derivative of a function indicates the slope of the tangent line to the curve of that function at any point. It provides valuable information about the function's rate of change and behavior.
In our exercise, the first derivative is:\[ f'(t) = 2t \]This tells us how \[ f(t) = t^2 - \frac{1}{2} \]'s value changes as \[ t \] changes. If \[ t \] increases, the slope \[ 2t \] increases, meaning the function is getting steeper. Conversely, if \[ t \] decreases, the slope decreases, and the function flattens out.
Understanding the first derivative helps us analyze the rate at which quantities change and predict future values based on their current rates of change. This concept is widely used in physics, engineering, economics, and other fields relying on change and motion analysis.