Power Rule for Derivatives: Examples & Explanation

The power rule for derivatives is simply a quick and easy rule that helps you find the derivative of certain kinds of functions. In this lesson, you will learn the rule and view a variety of examples.
What Is the Power Rule?
The power rule in calculus is a fairly simple rule that helps you find the derivative of a variable raised to a power, such as: x^5, 2x^8, 3x^(-3) or 5x^(1/2). All you do is take the exponent, multiply it by the coefficient (the number in front of the x), and decrease the exponent by 1.

A Few Examples
Let’s take a look at a few examples of the power rule in action.

Example 1
Our first example is y = 7x^5

Identify the power: 5

Multiply it by the coefficient: 5 x 7 = 35

Reduce the power by one: 4

You get dy / dx = 35x^4

Example 2
Here’s another example: y = 12x^2

y = (2 x 12) x^(2-1) = 24x

Example 3
And our next example: y = x^1000

y = 1000x^999

The previous three examples have used positive integer exponents. The same rule works if your exponents are negative or fractional.

Example 4
Here’s an example: y = 36x^(1/2)

y = (1/2)(36)x^(1/2 – 1) = 18x^(-1/2)

Example 5
Another example: y = 2x^(-3)

y = (-3)(2)x^(-3-1) = -6x^(-4)

Remember, in cases like this example, that one less than a negative number is a number even farther from zero. For example, one less than -3 is -4.

Working With Expressions Other Than Monomials
So far, you’ve just looked at monomials, expressions with only one term, like 5x. In algebra, you often encounter binomials, expressions with two terms, like 5x^4 + 2x, or trinomials, expressions with three terms, like 3x^2 – 2x + 6.

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