The power rule states that for a function $f(x) = ax^n$, where $a$ is a constant, the derivative is:
$f'(x) = anx^{n-1}$
This rule forms the basis for differentiating many types of functions, including polynomials and radicals.
The power rule works for any exponent, including negative integers, fractions, and zero. However, in the context of polynomials, we typically deal with non-negative integer exponents.
To apply the power rule, follow these steps:
Let's differentiate $f(x) = 3x^4$:
Therefore, $f'(x) = 12x^3$
When dealing with polynomials that have multiple terms, we apply the power rule to each term individually and then combine the results. This is possible due to the sum rule of differentiation, which states that the derivative of a sum is the sum of the derivatives.
For a polynomial function of the form:
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