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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.

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Example 1

Our first example is y = 7x^5Identify the power: 5Multiply it by the coefficient: 5 x 7 = 35Reduce the power by one: 4You get dy / dx = 35x^4

Example 2

Here’s another example: y = 12x^2y = (2 x 12) x^(2-1) = 24x

Example 3

And our next example: y = x^1000y = 1000x^999The 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.

The power rule works on binomials, trinomials or bigger nomials than that. You simply apply the power rule to each piece. The key is that each term can be written so that the variable is raised to a power. For example, 1 / x can be rewritten as x^(-1), but there is no way to rewrite x / (x+1) as a variable raised to a power.Let’s look at some examples working with binomials and trinomials. To do these, you will first rewrite the equation so each term is raised to a power and not in the denominator of a fraction. Then apply the power rule to each term.

Example 1

y = 3x^3 – 2x^2 + xy = 9x^2 – 4x + 1

Example 2

y = 1 / (x^2) + 1 / xy = x^(-2) + x^(-1)y = -2x^(-3) – x^(-2)

Lesson Summary

The power rule is a quick tool for finding the derivative of a function. It works whenever you can write the expression so that each term is simply a variable raised to a power. The power rule works if the exponent is negative or fractional as well.

It is one of the most commonly used techniques in calculus.

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