Explaining the Power Rule | Differentiating Functions of the Form f(x) = x^n

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 a 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 a real number. It states that the derivative of a power function f(x) = x^n is given by:

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

In simpler terms, if we have a function with a variable raised to a constant power, the derivative will be obtained by multiplying the original power by the variable, and then reducing the power by 1.

Here are a few examples of applying the power rule:

1. f(x) = x^2 -> f'(x) = 2x^(2-1) = 2x
This means that the derivative of the function f(x) = x^2 is f'(x) = 2x.

2. g(x) = x^5 -> g'(x) = 5x^(5-1) = 5x^4
The derivative of g(x) = x^5 is g'(x) = 5x^4.

3. h(x) = x^-3 -> h'(x) = -3x^(-3-1) = -3x^-4
The derivative of h(x) = x^-3 is h'(x) = -3x^-4.

It is important to note that the power rule can only be applied when differentiating functions where the power is a constant value and not a variable. If you have a function with a variable exponent, you would use a different rule, such as the exponential or logarithmic rule.

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