Chain Rule

How would you differentiate ? Expanding it gives six terms and a lot of arithmetic. And for or , expanding isn’t even an option. These are composite functions, one function inside another, and the chain rule handles all of them with the same short recipe. It’s the rule you’ll use more than any other for the rest of calculus.

What Is the Chain Rule?

If , where is differentiable at and is differentiable at , then

In words: differentiate the outside function, leaving the inside exactly as it is, then multiply by the derivative of the inside. Writing the inside as gives the Leibniz form, which is easier to remember:

To use it you need to see the function as an outer layer wrapped around an inner one. That decomposition skill comes from function composition in algebra: is “raise to the fifth power” applied to .

Chain Rule Explanation: Why Rates Multiply

Think of gears. If the middle gear turns 3 times for every turn of the first, and the last gear turns 2 times for every turn of the middle one, then the last gear turns times for every turn of the first. Rates along a chain multiply.

A composite function is exactly that kind of chain. Changing changes the inside , at a rate . Changing changes , at a rate . So changes with at the product of the two rates. That’s the whole idea; the formula is just bookkeeping.

The picture shows it on a real graph. goes through its cycle twice as fast as , because its inside, , moves twice as fast as . So it’s twice as steep at every matching point: its derivative is , the outer derivative times the inner derivative 2.

How to Use the Chain Rule

1. Split the function into an outer function and an inner one. Ask: “what’s the last thing I’d do if I were calculating this?” That’s the outer layer.
2. Differentiate the outer function, keeping the inside untouched in its slot.
3. Differentiate the inner function.
4. Multiply the two results.

The most common mistake by far is stopping after step 2. The inner derivative is not optional: forgetting it is like saying the last gear turns twice per turn of the first gear because it turns twice per turn of the middle one.

The Chain Rule and the Power Rule Together

The most frequent use of the chain rule is a whole expression raised to a power. Combining it with the power rule gives the general power rule:

Bring the exponent down, lower it by one, keep the inside, then multiply by the derivative of the inside. For example, . Roots and fractions work too, once they’re rewritten as powers.

Example 1: A Power of a Linear Expression

Differentiate .

Outer: , derivative . Inner: , derivative 3.

Example 2: A Square Root

Differentiate .

Write it as . The outer derivative is and the inner derivative is :

Example 3: A Trig Function With a Non-Trivial Inside

Differentiate .

The outer function is sine, whose derivative is cosine; the inside stays put:

Compare , where the outer function is the square: its derivative is . Where the brackets go decides which layer is outside.

Example 4: Exponentials

Differentiate and .

Since , the exponential stays the same and gets multiplied by the derivative of its exponent:

Example 5: One Over a Power

Differentiate .

Rewrite as instead of reaching for the quotient rule:

Example 6: The Chain Rule With Three Functions

Differentiate .

Using the chain rule is like peeling an onion: work from the outside in, one layer at a time. Here there are three layers, the cube, the sine and :

The rule for any number of layers is the same: one derivative per layer, each with everything inside it left alone, all multiplied together.

Example 7: The Chain Rule With the Product Rule

Differentiate .

This is a product, so start with the product rule. The second factor needs the chain rule, but its inner derivative is just 1:

Factoring out the common is worth the effort: it shows the slope is zero at , and .

Example 8: Using a Table of Values

Suppose , and . Find the derivative of at .

The key step is that is evaluated at , not at 1. Exam tables often include a tempting value to catch exactly this.

Example 9: The Chain Rule With Logs

Differentiate .

Using :

For any , the derivative is the derivative of the something divided by the something itself.

Why the Chain Rule Works

The derivation of the chain rule starts from the average rate of change. If a small change causes a change in the inside, which causes a change in the outside, then

As , as well, because a differentiable function is continuous. The two fractions become and . (A fully rigorous proof also handles the case where is zero for small , but the idea is exactly this cancellation, which is why Leibniz notation makes the rule look so natural.)

Common Mistakes With the Chain Rule

• Forgetting the inner derivative. is , not .
• Changing the inside. The outer derivative keeps the inside as it is: , not .
• Stopping one layer early with three or more layers. Count the layers and make sure you have that many factors.
• Evaluating at the wrong point with tables: it’s , not .
• Mixing up and . The first is a square of a sine; the second is a sine of a square.

Practice Problems

Try each one before opening the answer. The practice set has 12 more with instant checking.

Problem 1. Differentiate .

Problem 2. Differentiate .

Problem 3. Differentiate .

Problem 4. Differentiate .

Problem 5. If , and , find the derivative of at .

With the power, product, quotient and chain rules in hand, you can differentiate almost any formula built from powers. Next come the derivatives of trig functions, where the chain rule gets put to work straight away.