Mastering the Power Rule | Understanding and Applying the Fundamental Rule of Differentiation in Calculus

d/dx [x^n] (Power Rule)

The power rule is a fundamental rule of differentiation in calculus that allows us to find the derivative of a power function, where the variable is raised to a constant power

The power rule is a fundamental rule of differentiation in calculus that allows us to find the derivative of a power function, where the variable is raised to a constant power. In particular, it states that the derivative of x raised to the power of n, denoted as x^n, is equal to n times x raised to the power of n-1.

Mathematically, the power rule can be expressed as:

d/dx [x^n] = n * x^(n-1)

Let’s break this down and understand how it works. Consider a power function x^n, where n is a constant. To find its derivative with respect to x, we calculate d/dx [x^n]. The derivative indicates how the function is changing as x varies.

Using the power rule, we differentiate each term of the function separately. For the term x, the derivative is simply 1 because the derivative of x with respect to x is 1. For the term n, which is a constant, its derivative is 0 as it is not changing with respect to x.

The only term that remains is x^(n-1). Applying the power rule to this term, we multiply its exponent (n-1) by the coefficient in front of x, which is 1. This gives us the derivative of x^n as n * x^(n-1).

For example, let’s find the derivative of the function f(x) = x^3:

Using the power rule, we have:

d/dx [x^3] = 3 * x^(3-1)
= 3 * x^2

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

The power rule is an essential tool in calculus, as it allows us to differentiate various types of functions involving powers. By understanding this rule and practicing its application, we can derive the derivatives of many functions more efficiently and solve a wide range of mathematical problems.

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