Differentiation Rules

This purely reference page lists some of the most important differentiation rules, including constant rule, power rule, sum rule, difference rule, product rule, quotient rule, and chain rule.

You can access this page from the Key Formulas menu item at the top of every page.

Constant Rule

The derivative of a constant function is 0.

$𝑓 (𝑥) = 𝑐 \Rightarrow 𝑓' (𝑥) = 0$

Power Rule

The derivative of $𝑥 𝑛$ is $𝑛 𝑥 𝑛 - 1$ .

$𝑓 (𝑥) = 𝑥 𝑛 \Rightarrow 𝑓' (𝑥) = 𝑛 𝑥 𝑛 - 1$

To practice using the rules above, visit our free Derivatives of Constant, Linear, & Power Functions page.

Sum Rule

The derivative of the sum of two functions is the sum of the derivatives of the two functions.

$𝑓 (𝑥) = 𝑔 (𝑥) + ℎ (𝑥) \Rightarrow 𝑓' (𝑥) = 𝑔' (𝑥) + ℎ' (𝑥)$

Difference Rule

The derivative of the difference of two functions is the difference of the derivatives of the two functions.

$𝑓 (𝑥) = 𝑔 (𝑥) - ℎ (𝑥) \Rightarrow 𝑓' (𝑥) = 𝑔' (𝑥) - ℎ' (𝑥)$

Product Rule

The derivative of the product of two functions is [the first function times the derivative of the second function] plus [the second function times the derivative of the first function].

𝑓 (𝑥) = 𝑔 (𝑥) ℎ (𝑥) \Rightarrow

𝑓' (𝑥) = 𝑔 (𝑥) ℎ' (𝑥) + ℎ (𝑥) 𝑔' (𝑥) = [(d e r i v o f t h e 1 s t) \times (t h e 2 n d)] + [(t h e 1 s t) \times (d e r i v o f t h e 2 n d)]

To practice using the Product Rule, visit our free Product Rule page.

Quotient Rule

The derivative of the quotient of two functions is [the derivative of the numerator times the denominator] minus [the numerator times the derivative of the denominator], all divided by [the square of the denominator].

$𝑓 (𝑥) = 𝑔 (𝑥) ℎ (𝑥) \Rightarrow 𝑓' (𝑥) = ℎ (𝑥) 𝑔' (𝑥) - 𝑔 (𝑥) ℎ' (𝑥) ℎ (𝑥) 2 = [(d e r i v o f t h e n u m e r a t o r) \times (t h e d e n o m i n a t o r)] - [(t h e n u m e r a t o r) \times (d e r i v o f t h e d e n o m i n a t o r)] a l l d i v i d e d b y [t h e d e n o m i n a t o r, s q u a r e d]$

Many students remember the Quotient Rule by thinking of the numerator as "hi," the demoninator as "lo," the derivative as "d," and then singing

"lo d-hi minus hi d-lo over lo-lo"

To practice using the Quotient Rule, visit our free Quotient Rule page.

Chain Rule

The derivative of the composition of two functions is [the derivative of the outer function times the derivative of the inner function].

𝑓 (𝑥) = 𝑔 (ℎ (𝑥)) \Rightarrow

$𝑓' (𝑥) = 𝑔' (ℎ (𝑥)) \cdot ℎ' (𝑥) = [[d e r i v o f o u t e r f u n c t i o n, e v a l u a t e d a t i n n e r f u n c t i o n] \times [d e r i v o f i n n e r f u n c t i o n]]$

Alternatively, if we write $𝑦 = 𝑓 (𝑢)$ and $𝑢 = 𝑔 (𝑥),$ then

𝑑 𝑦 𝑑 𝑥 = 𝑑 𝑦 𝑑 𝑢 \cdot 𝑑 𝑢 𝑑 𝑥

Informally:

𝑑 𝑓 𝑑 𝑥 = [𝑑 𝑓 𝑑 (s t u f f)] \times 𝑑 𝑑 𝑥 (s t u f f)

For many examples and practice problems with the Chain Rule, visit our free Basic Practice with the Chain Rule and Chain Rule - Deeper Work pages.

One quick example of the Chain Rule

Consider $𝑓 (𝑥) = (𝑥 2 + 1) 7 .$

To find the derivative, think something like: "The function is a bunch of stuff to the 7th power. So the derivative is 7 times that same stuff to the 6th power, times the derivative of that stuff."

𝑓 (𝑥) = (s t u f f) 7; s t u f f = 𝑥 2 + 1 T h e n 𝑓 (𝑥) = 𝑑 𝑓 𝑑 𝑥 = 7 (s t u f f) 6 \cdot (𝑑 𝑑 𝑥 (𝑥 2 + 1)) = 7 (𝑥 2 + 1) 6 \cdot (2 𝑥) ✓

Note: You'd never actually write out "stuff = ....".
Instead just hold in your head what that "stuff" is, and proceed to write down the required derivatives. With a little practice, this becomes quite routine. Indeed, using our free practice problems you'll quickly come to recognize the answer to some Chain Rule multiple choice questions without even writing anything down!

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Tip: You can differentiate any function, for free, using WolframAlpha's Online Derivative Calculator.