How to Take a Derivative with the Power Rule

The derivative of a function tells you its rate of change, the slope of its graph at each point. For terms like xⁿ, there is a quick shortcut called the power rule.

Try it with your own problem ↓

The steps

Power rule: the derivative of xⁿ is n · xⁿ⁻¹. Bring the exponent down in front, then subtract 1 from the exponent.
A number in front stays and is multiplied: the derivative of a · xⁿ is a · n · xⁿ⁻¹.
The derivative of x (that is, x¹) is 1, and the derivative of any plain number is 0.
For a sum or difference, take the derivative of each term separately.

A worked example

Find the derivative of f(x) = 4x³ − 5x² + 7x − 9.

4x³: bring down the 3 and multiply by 4 to get 12, and subtract 1 from the exponent: 12x².
−5x²: 2 × −5 = −10, and the exponent becomes 1: −10x.
7x: the derivative of x is 1, so 7 × 1 = 7.
−9: a plain number, so the derivative is 0.
Put it together: f′(x) = 12x² − 10x + 7.

f′(x) = 12x² − 10x + 7.

Try it with your own problem

Open the free The Rate Machine with this example already filled in. It shows every step, and you can type over it with your own numbers.

Work it out: derivative of 4x^3 - 5x^2 + 7x - 9Opens The Rate Machine with this example

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Common mistakes

Forgetting to lower the exponent. The derivative of x³ is 3x², not 3x³.
Forgetting the coefficient. The derivative of 4x³ is 12x², not 3x².
Keeping the constant. A plain number has derivative 0.
Using the power rule on 2ˣ. That is a different kind of function. The rule is only for xⁿ.

Questions people ask

What does the derivative mean?

The slope of the tangent line at a point, or how fast a quantity is changing. If f is position, f′ is velocity.

How do I find the slope at a specific point?

Put the x-value into the derivative. For f′(x) = 12x² − 10x + 7 at x = 1: 12 − 10 + 7 = 9.

What if the function is more complicated?

There are more rules: the product, quotient and chain rules. The calculus helper handles those too.

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