The Chain Rule

You already use chain-rule thinking every day, even if you have never heard the term. When you check the weather and think, “It’s getting colder because the sun is setting,” you are linking two rates of change: temperature depends on sunlight, and sunlight depends on time. The chain rule is simply the mathematical machinery for connecting these linked rates. Once you see the pattern, you will wonder how you ever managed without it.

This lesson will show you how to differentiate composite functions—functions built by plugging one function inside another. The chain rule is arguably the most powerful differentiation technique you will learn, because real-world quantities almost always depend on other quantities in layered ways. Master this rule, and you unlock the ability to handle exponentials, trigonometric expressions, nested powers, and far more.

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Core Concepts

What Is a Composite Function?

Before diving into derivatives, let us make sure we are comfortable with composite functions themselves. A composite function is what you get when you feed the output of one function into the input of another.

Suppose you have two functions:

• $g(x) = x^2$ (the “inner” function)
• $f(u) = \sin(u)$ (the “outer” function)

The composite function $(f \circ g)(x)$ means “first apply $g$, then apply $f$”:

$$(f \circ g)(x) = f(g(x)) = f(x^2) = \sin(x^2)$$

Notice the order carefully: $g$ acts first (it is “inside”), and $f$ acts on the result (it is “outside”). Think of it like putting on socks and then shoes—the socks go on first, but the shoes are what the world sees on the outside.

The Problem: How Do We Differentiate Composites?

You already know how to differentiate $\sin(x)$ and $x^2$ separately. But what about $\sin(x^2)$? You cannot simply differentiate the outer function and ignore what is inside. The derivative of $\sin(x^2)$ is not $\cos(x^2)$.

Here is the intuition: when you differentiate $\sin(u)$ with respect to $u$, you get $\cos(u)$. But if $u$ itself is changing (because $u = x^2$ depends on $x$), then you need to account for how fast $u$ is changing too. The chain rule tells you exactly how to do this.

The Chain Rule: Two Equivalent Forms

Function notation: $$\frac{d}{dx}[f(g(x))] = f’(g(x)) \cdot g’(x)$$

In words: differentiate the outer function (keeping the inner function unchanged), then multiply by the derivative of the inner function.

Leibniz notation: $$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$$

This version looks like the $du$’s “cancel,” and while that is not quite rigorous, it is an excellent memory aid. If $y$ depends on $u$, and $u$ depends on $x$, then the rate of change of $y$ with respect to $x$ is the product of the two individual rates.

Identifying Outer and Inner Functions

The key skill in applying the chain rule is recognizing the composite structure. Ask yourself: “What is the outermost operation, and what is its input?”

Example: For $y = (3x + 7)^5$:

• The outermost operation is “raise something to the 5th power”
• The “something” (the inner function) is $3x + 7$
• So $f(u) = u^5$ and $g(x) = 3x + 7$

Example: For $y = e^{x^2}$:

• The outermost operation is “take $e$ to a power”
• The exponent (the inner function) is $x^2$
• So $f(u) = e^u$ and $g(x) = x^2$

Example: For $y = \sqrt{\sin(x)}$:

• The outermost operation is “take the square root”
• The thing inside the square root is $\sin(x)$
• So $f(u) = \sqrt{u}$ and $g(x) = \sin(x)$

A Step-by-Step Process

Here is a reliable method for applying the chain rule:

1. Identify the composite structure. Find the outer function $f$ and the inner function $g(x)$.
2. Differentiate the outer function as if the inner part were just a single variable. This gives you $f’(g(x))$.
3. Multiply by the derivative of the inner function. Compute $g’(x)$ and multiply.
4. Simplify if desired.

Let us apply this to $y = \sin(x^2)$:

1. Outer: $f(u) = \sin(u)$. Inner: $g(x) = x^2$.
2. $f’(u) = \cos(u)$, so $f’(g(x)) = \cos(x^2)$.
3. $g’(x) = 2x$.
4. Answer: $\frac{dy}{dx} = \cos(x^2) \cdot 2x = 2x\cos(x^2)$.

Combining the Chain Rule with Other Rules

Real problems often require using the chain rule alongside the product rule or quotient rule. The key is to work systematically from the outside in.

Chain Rule + Product Rule:

If $y = f(x) \cdot g(x)$ where $f$ or $g$ (or both) require the chain rule, use the product rule first, then apply the chain rule to each piece.

Chain Rule + Quotient Rule:

Similarly, if $y = \frac{f(x)}{g(x)}$ and the numerator or denominator involves a composition, use the quotient rule for the overall structure, then the chain rule for the individual pieces.

Nested Compositions: Multiple Chain Rules

Sometimes you have a function inside a function inside a function. For example: $$y = \sin(\cos(x^2))$$

Here you have three layers:

• Outermost: $\sin(\square)$
• Middle: $\cos(\square)$
• Innermost: $x^2$

You apply the chain rule repeatedly, working from the outside in:

$$\frac{dy}{dx} = \cos(\cos(x^2)) \cdot \frac{d}{dx}[\cos(x^2)]$$

Now apply the chain rule again to $\frac{d}{dx}[\cos(x^2)]$:

$$= \cos(\cos(x^2)) \cdot (-\sin(x^2)) \cdot 2x$$

$$= -2x \sin(x^2) \cos(\cos(x^2))$$

The pattern extends: each time you “peel off” one layer, you multiply by the derivative of what remains inside.

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Notation and Terminology

Term	Meaning	Example
Composite function	A function formed by applying one function to the output of another	$\sin(x^2)$, where $\sin$ is applied to the output of $x^2$
Outer function	The “wrapper” function—the last operation performed	In $\sin(x^2)$, the outer function is $\sin(\square)$
Inner function	The “filling”—the function whose output becomes the input	In $\sin(x^2)$, the inner function is $x^2$
Chain rule	The rule $(f \circ g)’(x) = f’(g(x)) \cdot g’(x)$ for differentiating composites
Leibniz form	$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$	If $y = u^3$ and $u = 2x + 1$, then $\frac{dy}{dx} = 3u^2 \cdot 2 = 6(2x+1)^2$
Nested composition	A composite of three or more functions	$e^{\sin(x^2)}$ has three layers

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Examples

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Key Properties and Rules

The Chain Rule Formula (Both Forms)

Prime notation: $$[f(g(x))]’ = f’(g(x)) \cdot g’(x)$$

Leibniz notation: $$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$$

Common Patterns to Memorize

These patterns come up so often that it is worth knowing them by heart:

Function	Derivative
$[g(x)]^n$	$n[g(x)]^{n-1} \cdot g’(x)$
$e^{g(x)}$	$e^{g(x)} \cdot g’(x)$
$\ln(g(x))$	$\frac{g’(x)}{g(x)}$
$\sin(g(x))$	$\cos(g(x)) \cdot g’(x)$
$\cos(g(x))$	$-\sin(g(x)) \cdot g’(x)$
$\tan(g(x))$	$\sec^2(g(x)) \cdot g’(x)$

The “Outside-In” Principle

When differentiating a composite function:

1. Start with the outermost function
2. Differentiate it, leaving the inside alone
3. Multiply by the derivative of the next layer in
4. Repeat until you reach the innermost function

Combining Rules: Order of Operations

When a problem requires multiple differentiation rules:

• Product of composites: Product rule first, chain rule for each factor
• Quotient of composites: Quotient rule first, chain rule for numerator and denominator
• Composition of products: Chain rule first, product rule for the inner function

The Chain Rule Extends

For a composition of $n$ functions: $$\frac{d}{dx}[f_1(f_2(f_3(\cdots f_n(x))))] = f_1’(f_2(\cdots)) \cdot f_2’(f_3(\cdots)) \cdot \cdots \cdot f_n’(x)$$

Each derivative is evaluated at the appropriate composition.

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Real-World Applications

Temperature and Altitude

Imagine you are hiking up a mountain. The temperature $T$ depends on your altitude $h$, and your altitude depends on the time $t$ you have been hiking. If you want to know how fast the temperature is changing with respect to time, you need the chain rule:

$$\frac{dT}{dt} = \frac{dT}{dh} \cdot \frac{dh}{dt}$$

For instance, if temperature drops by $2°C$ per 100 meters of altitude gain, and you are climbing at 50 meters per minute:

$$\frac{dT}{dt} = \left(-\frac{2°C}{100 \text{ m}}\right) \cdot \left(\frac{50 \text{ m}}{\text{min}}\right) = -1°C/\text{min}$$

The temperature you feel is dropping at 1 degree per minute.

Gear Ratios and Mechanical Systems

In a car’s transmission, engine speed and wheel speed are linked through gear ratios. If the engine spins at rate $\omega_e$ and the gear ratio is $r$, then wheel speed is $\omega_w = r \cdot \omega_e$. The rate of change of wheel speed with respect to time involves the chain rule:

$$\frac{d\omega_w}{dt} = r \cdot \frac{d\omega_e}{dt}$$

This is why shifting to a lower gear (higher $r$) gives you more acceleration—the same engine acceleration produces a larger wheel acceleration.

Population Dynamics

In ecology, a predator population $P$ might depend on a prey population $N$, which in turn depends on time $t$ as seasons change. The rate of change of the predator population with respect to time is:

$$\frac{dP}{dt} = \frac{dP}{dN} \cdot \frac{dN}{dt}$$

This captures how changes in prey availability (due to seasonal fluctuations) propagate through to predator populations.

Unit Conversions and Rate Changes

Suppose you know how a quantity changes with respect to one unit, and you want to convert to another unit. For example, if you know fuel efficiency in miles per gallon and want to find it in kilometers per liter:

The chain rule underlies the conversion: $\frac{d(\text{km})}{d(\text{L})} = \frac{d(\text{km})}{d(\text{mi})} \cdot \frac{d(\text{mi})}{d(\text{gal})} \cdot \frac{d(\text{gal})}{d(\text{L})}$

Financial Models

In economics, profit $P$ might depend on quantity sold $Q$, which depends on price $p$, which depends on time $t$ (due to inflation or market conditions). The chain rule connects all these rates:

$$\frac{dP}{dt} = \frac{dP}{dQ} \cdot \frac{dQ}{dp} \cdot \frac{dp}{dt}$$

This helps businesses understand how time-dependent market changes affect their bottom line.

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Self-Test Problems

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Summary

• Composite functions are built by applying one function to the output of another: $(f \circ g)(x) = f(g(x))$.

• The chain rule tells us how to differentiate composites: $$\frac{d}{dx}[f(g(x))] = f’(g(x)) \cdot g’(x)$$

• In Leibniz notation, the chain rule looks like: $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$.

• To apply the chain rule, identify the outer and inner functions. The outer function is the last operation performed; the inner function is its input.

• Differentiate from outside to inside: first differentiate the outer function (keeping the inner unchanged), then multiply by the derivative of the inner function.

• The chain rule combines with other rules. For products or quotients of compositions, use the product or quotient rule first, then apply the chain rule to each piece.

• For nested compositions (three or more layers), apply the chain rule repeatedly—you get one factor for each layer.

• Common patterns like $[g(x)]^n$, $e^{g(x)}$, and $\ln(g(x))$ appear frequently. Recognizing them speeds up your work.

• The chain rule has countless applications: physics (related rates), engineering (gear systems), biology (population dynamics), economics (linked variables), and anywhere quantities depend on each other in layers.

• The chain rule is often called the most important differentiation rule because most real-world functions are compositions. Once you master it, you can differentiate almost anything.