Answer
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Work Step by Step
These three formulas, known as the power rule, constant multiple rule, and sum rule respectively, are fundamental rules in calculus that enable us to differentiate any polynomial function. Here's how they work:
a. Power Rule:
The formula \( \frac{d}{d x}\left(x^{n}\right)=n x^{n-1} \) allows us to find the derivative of any term in the form \( x^n \), where \( n \) is a constant. This rule states that to differentiate a term with a variable raised to a constant power, you bring down the power as a coefficient and reduce the power by 1.
b. Constant Multiple Rule:
The formula \( \frac{d}{d x}(c u)=c \frac{d u}{d x} \) tells us that when we differentiate a constant multiple of a function, we can pull the constant multiple out of the derivative. This means that the derivative of a constant times a function is equal to the constant times the derivative of the function.
c. Sum Rule:
The formula \( \frac{d}{d x}\left(u_{1}+u_{2}+\cdots+u_{n}\right)=\frac{d u_{1}}{d x}+\frac{d u_{2}}{d x}+\cdots+\frac{d u_{n}}{d x} \) states that the derivative of a sum of functions is equal to the sum of the derivatives of those functions. In other words, you can differentiate each term in the polynomial separately and then sum up the results.
By using these three rules, we can differentiate any polynomial function because polynomials are composed of terms that are either constants, powers of \( x \), or sums of these terms. Applying these rules iteratively to each term in the polynomial allows us to find the derivative of the entire polynomial function.
