The Power Rule | A Shortcut Method for Calculus Differentiation of Power Functions

Power Rule

The power rule is a fundamental rule in calculus that is used to differentiate functions of the form f(x) = x^n, where n is a constant

The power rule is a fundamental rule in calculus that is used to differentiate functions of the form f(x) = x^n, where n is a constant. In simple terms, it gives us a shortcut method to find the derivative of a power function.

According to the power rule, if we have a function f(x) = x^n, then its derivative, denoted as f'(x) or dy/dx, is given by:

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

Here, nx^(n-1) represents the product of the constant n and the base x raised to the power (n-1). The power rule holds true for any real number n, including positive integers, negative integers, fractions, and even irrational numbers.

For example, let’s consider the function f(x) = x^3. Using the power rule, we can find its derivative:

f'(x) = 3x^(3-1)
= 3x^2

So, the derivative of x^3 is 3x^2.

The power rule greatly simplifies the process of finding derivatives of power functions, allowing us to avoid more complicated methods such as using limits or the binomial theorem. It is a fundamental tool used in calculus to differentiate a wide variety of functions and is an important concept to understand in order to solve more complex mathematical problems.

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