Mastering the Power Rule: A Comprehensive Guide to Differentiating Polynomial Functions

Power Rule

The power rule is a fundamental rule in calculus that allows us to differentiate functions of the form f(x) = x^n, where n is any real number

The power rule is a fundamental rule in calculus that allows us to differentiate functions of the form f(x) = x^n, where n is any real number. This rule is particularly useful when dealing with polynomial functions.

The power rule states that if we have a function f(x) = x^n, then the derivative of f(x) with respect to x is given by:

f'(x) = n * x^(n-1)

Let’s break this down step by step:

1. Start with the function f(x) = x^n, where n is any real number.

2. To find the derivative, we differentiate the function with respect to x. This means we find the rate at which the function is changing at each point.

3. Applying the power rule, we multiply the exponent (n) by the coefficient in front of x, which is 1.

4. Then, we subtract 1 from the exponent.

5. Finally, we write the result as the derivative, f'(x).

Let’s look at a few examples to understand how the power rule works:

Example 1: f(x) = x^2
Using the power rule, we differentiate f(x) to find f'(x):
f'(x) = 2 * x^(2-1) = 2 * x^1 = 2x

So, the derivative of f(x) = x^2 is f'(x) = 2x.

Example 2: f(x) = x^3
Applying the power rule, we differentiate f(x):
f'(x) = 3 * x^(3-1) = 3 * x^2

Therefore, the derivative of f(x) = x^3 is f'(x) = 3x^2.

Example 3: f(x) = x^(-2)
Using the power rule, we differentiate f(x):
f'(x) = -2 * x^(-2-1) = -2 * x^(-3) = -2/x^3

So, the derivative of f(x) = x^(-2) is f'(x) = -2/x^3.

The power rule is a powerful tool in calculus that simplifies the process of finding derivatives of functions with exponents. It can be extended to more complex functions by applying other differentiation rules, such as the product rule or the chain rule, when necessary.

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