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Differentiation is a fundamental concept in calculus which involves finding the derivative of a function. The derivative represents the rate at which a function is changing at any given point. It is like asking "How fast is something happening?" or "How does one quantity change in relation to another?".To differentiate a function means to compute its derivative. This involves mathematical processes that apply rules and formulas to find how much a small change in the input of the function returns a change in its output. For example, if you have a function expressed as \[ f(x) = ax^n \]where \(a\) and \(n\) are constants, the derivative is found by using the power rule: \[ f'(x) = nax^{n-1} \]Here, you multiply the power of \(x\) by the coefficient in front and then decrease the power by one.Differentiating a function, much like with our original exercise, requires applying these rules to each term individually. The derivative of a sum of functions is the sum of their derivatives. This is because differentiation is a linear operation, which allows it to be applied term by term.

A constant function is one that returns the same value no matter what input it is given. Mathematically, it can be written as:\[ f(x) = c \]where \(c\) is a constant number.In the context of differentiation, finding the derivative of a constant function is straightforward. Since the value of the function does not change irrespective of the input, the rate of change is zero. That's why the derivative of a constant is always \(0\).In our original exercise, we had the constant function value \(2\) as part of \( g(x) = -5x + 2 \). When we differentiate this constant term, it simply contributes zero to the derivative. This reinforces the notion that constant functions have a zero rate of change across their domain.

A linear function is one of the simplest forms of functions in mathematics, typically expressed in the form:\[ f(x) = ax + b \]Here, \(a\) and \(b\) are constants, with \(a\) representing the slope and \(b\) representing the y-intercept of the graph of the function.The derivative of a linear function is simply its slope, \(a\). This is because the "rate of change" or "slope" is constant across the entire line. No matter what value \(x\) takes, the change in \(f(x)\) as \(x\) increases is always \(a\).In our exercise, the linear function \(-5x\) has a slope of \(-5\). Differentiating it uses the basic rule that the derivative of \(ax\) is \(a\). So, the derivative of \(-5x + 2\) is simply \(-5\), confirming the linear nature of the term and its constant rate of change.