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Differentiation is a fundamental operation in calculus used to compute the rate at which a function is changing at any given point. It is a cornerstone of calculus that involves finding the derivative of a function. The derivative of a function at a point can be geometrically interpreted as the slope of the tangent line to the function's graph at that point. Algebraically, it provides the rate of change of a function's output with respect to a change in the input.

For example, in physics, differentiation is used to determine the velocity (which is the derivative of position with respect to time) or acceleration (the derivative of velocity). In our current exercise, we compute the derivative of a product of two functions, which requires knowledge of rules that ease the differentiation process.

The chain rule is an essential rule in differentiation that deals with the derivative of composite functions. A composite function is created when one function is nested inside another, such as in the expression \( g(f(x)) \). If you need to find the derivative of \( h(x) = g(f(x)) \), the chain rule states that you should multiply the derivative of the outer function evaluated at the inner function by the derivative of the inner function. In notation, it is expressed as \( h'(x) = g'(f(x))\cdot f'(x) \).

While the exercise provided doesn't directly involve composite functions, understanding the chain rule can help when dealing with more complex products or functions that are not merely multiplied by one another, but where one function's output is the argument of the other.

The power rule is a quick and useful technique to find the derivative of functions that are of the form \( x^n \) where \( n \) is any real number. According to the power rule, the derivative of \( x^n \) is \( nx^{n-1} \). It simplifies the process of differentiation as it eliminates the need for limit procedures every time you need to find the derivative of a power function.

In the context of the exercise, the power rule is applied to find the derivatives of \( x^3 \) and to simplify the final expression after applying the product rule. By using the power rule, we can quickly determine the derivatives to be \( 3x^2 \) and \( 3x^2-5 \), as each term in the functions \( f(x) \) and \( g(x) \) is a power function.