Applying the Power Rule

What the Power Rule Is For

The power rule is the basic derivative rule for functions whose variable is raised to a constant power. It is one of the first rules you should recognize when taking derivatives and forms the foundation for differentiating polynomials and many algebraic functions.

The key idea is simple: bring the exponent down as a coefficient, then reduce the exponent by 1.

The Formula

For any real constant exponent :

In words: multiply by the exponent and subtract 1 from the exponent.

For example, if you see , the exponent is 5. Bring the 5 down in front and change the exponent from 5 to 4.

A Step-by-Step Method

Identify the exponent. Find the power on the variable.
Bring the exponent down. It becomes the coefficient of the derivative.
Subtract 1 from the exponent.
Simplify. Combine constants or rewrite the result if necessary.

Worked Examples

Example 1: A Basic Power

Differentiate .

Step 1: The exponent is .

Step 2: Bring the exponent down:

Step 3: Reduce the exponent by 1:

Result:

Example 2: A Polynomial Term

Differentiate .

Step 1: Apply the power rule to the first term:

Step 2: Apply the power rule to the second term:

Step 3: The derivative of a constant is 0:

Result:

Example 3: Negative Exponents

Differentiate .

Step 1: Bring the exponent down:

Step 2: Reduce the exponent by 1:

Step 3: Simplify the coefficient:

Result:

Example 4: Fractional and Radical Powers

Differentiate .

Step 1: Rewrite the square root as a fractional exponent:

Step 2: Apply the power rule:

Step 3: Rewrite the result in radical form if desired:

Result:

Example 5: A Coefficient in Front

Differentiate .

Step 1: Keep the constant coefficient and apply the power rule to .

Step 2:

Step 3: Multiply the constants:

Result:

When the Power Rule Works Directly

The power rule applies directly when the base is simply and the exponent is a constant. Examples include , , and .

However, be careful when something more complicated is inside the power. For example, is not handled by the basic power rule alone. It requires the chain rule because the base is , not simply .

Power Rule + Other Rules

The power rule frequently appears together with other differentiation rules. On the AP exam, you may need to recognize which rule applies to each part of an expression.

Example 6: Power Rule with the Product Rule

Differentiate .

Step 1: This is a product, so use the product rule:

Step 2: Differentiate using the power rule:

Step 3: Differentiate :

Result:

Common Mistakes

Forgetting to reduce the exponent

The exponent must decrease by 1. For , the derivative is not . It is .

Forgetting the coefficient

The original exponent becomes the new coefficient. For , you must bring the 7 down: .

Missing negative exponents

The power rule works for negative exponents too. If , then the derivative is .

Using the power rule when a chain rule is required

For , the base is not simply . You must multiply by the derivative of the inside expression.

A Useful Mental Shortcut

When you see a simple power of , think:

For example:

becomes
becomes
becomes
becomes

Practice Problems

Question 1

Differentiate .

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Question 2

Differentiate .

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Question 3

Differentiate .

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Question 4 (Fractional Exponent)

Differentiate .

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Question 5 (Negative Exponent)

Differentiate .

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Question 6 (Combined Rules)

Differentiate .

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