Understanding the Power Rule: A Comprehensive Guide to Differentiating Functions with Powers of Variables

Power Rule

The power rule is a fundamental rule in calculus that allows us to differentiate functions involving powers of a variable

The power rule is a fundamental rule in calculus that allows us to differentiate functions involving powers of a variable. It states that if we have a function of the form f(x) = x^n, where n is any real number, then the derivative of this function is given by:

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

Let’s look at a few examples to understand how the power rule works in practice.

Example 1:
Consider the function f(x) = x^3. To find its derivative, we can use the power rule: f'(x) = 3*x^(3-1) = 3*x^2. So, the derivative of f(x) = x^3 is f'(x) = 3x^2.

Example 2:
Now, let’s consider a function with a negative power. Suppose we have g(x) = x^(-2). Applying the power rule, we find g'(x) = -2*x^(-2-1) = -2*x^(-3). Therefore, the derivative of g(x) = x^(-2) is g'(x) = -2/x^3.

Example 3:
Let’s explore the power rule with a fractional power. Suppose h(x) = x^(1/2) = √x. In this case, we still apply the power rule: h'(x) = (1/2)*x^((1/2)-1) = (1/2)*x^(-1/2). So, the derivative of h(x) = √x is h'(x) = (1/2)/√x = 1/(2√x).

These examples illustrate the power rule in action. It allows us to find the derivative of a function involving powers of x by simply multiplying the exponent with the coefficient and reducing the exponent by 1. This rule is especially useful when dealing with polynomial functions and various other mathematical expressions in calculus.

More Answers: