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A cubic function is a type of polynomial function that has the general form . In other words, it is a third-degree polynomial. These functions have the highest power of the variable as 3. They often exhibit a distinctive 'S' shape when graphed, and they can have up to three real roots (the values of x where the function equals zero).For instance, the function given in the exercise, , is a quintessential cubic function. This is because the highest exponent of x is 3. Generally, cubic functions can have one local maximum and one local minimum due to their shape. By understanding cubic functions, you can interpret various physical phenomena, such as motion and growth patterns.

The power rule is a fundamental principle in differentiation that is particularly useful for polynomial functions. It states that to differentiate a function , you multiply the exponent by the coefficient and then subtract one from the exponent. This can be formalized as: .For our function , we apply the power rule to find its derivative. For , the power rule gives us . By following this rule, we derive that the derivative of is . This tool is indispensable for calculating the rate of change in polynomials and many other functions.

Calculating derivatives allows us to understand the rate of change of a function at any given point. For the function in our example, , we already determined that its derivative is via the power rule.First, we compute the function value: . Substituting gives us = -125.Next, to find the rate of change at this point, we substitute into the derivative . This yields . After computing, we see that = 75, meaning the rate of change of our cubic function at is 75.