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When it comes to understanding calculus, differentiation of piecewise functions is a fundamental skill. A piecewise function is merely a function that has different expressions based on the interval of the input value, x. Differentiation of these functions involves taking the derivative of each piece separately, just as we would for standard functions.

For instance, if we have a function defined by different formulas in different intervals, we'd differentiate each formula independently within its respective interval. It's crucial to calculate these derivatives accurately because they play a pivotal role in determining the function's behavior and its continuity at specific points.
In our exercise example, the function \( f(x) \) is broken into two intervals: \( x \) less than or equal to 1, and \( x \) greater than 1. For each interval, we derived \( f'(x) \), which results in two separate expressions for the derivative. This differentiation is straightforward, but it's the subsequent step—ensuring the continuity of the derivative at the boundary between the pieces—that causes most of the challenges.

The continuity of a function's derivative is a deeper concept that often confuses students. Simply put, for a function to be smoothly continuous, not only the function itself but also its derivative must not have any abrupt changes or breaks.

In a mathematical context, a function's derivative is said to be continuous at a point if the left-hand limit and the right-hand limit at the point are equal. This means that for a piecewise function like the one in our exercise, we evaluate the derivative from both 'pieces' as we approach the boundary. If the two results match perfectly, the derivative is continuous, and the overall function is likely to be smooth across that region.
To achieve this continuity in our problem, we first found the derivatives separately for \( x \) less than and equal to 1 and for \( x \) greater than 1. Evaluating these at the point \( x=1 \) gave us two expressions. By setting them equal to each other, we ensured that the slopes on both sides of \( x=1 \) are the same, thus guaranteeing the smoothness of the graph at that point. This method is fundamental in calculus, as continuous derivatives usually imply a function is differentiable, and hence, much easier to work with analytically.

Calculus frequently involves finding unknown values that make a function behave in a certain way, especially when it comes to integrals, limits, and — as in our case — derivatives. When we're given a piecewise function with an unknown constant, the goal is to determine this constant so that the function meets the criteria specified in the problem, like being differentiable or continuous.

Determining constants in calculus is like solving a puzzle. We use conditions given by the problem (like continuity or a specific function value) to find an unknown constant. In our exercise, the condition was the continuity of the derivative at the boundary point of the two pieces of the piecewise function. By equating the derivatives from both intervals at the point and solving for the unknown constant \( a \), we found the value that makes the function's derivative continuous. \( a \) turned out to be 4 in this case.
This approach is not just mechanical; it involves a deep understanding of how functions and derivatives work. It's a testament to the beauty of calculus -- through a series of logical steps, we can deduce missing information that ensures the function maintains the desired behavior.