Chain Rule

🔗 The Chain Rule

Part 1 of 7 — Chain Rule Basics

Welcome to the Chain Rule — arguably the most important differentiation rule in calculus!

Why Do We Need the Chain Rule?

So far, you can differentiate functions like $x^3$, $\sin x$, or $e^x$. But what about composite functions — functions inside other functions?

• $(3x + 1)^5$ — expanding this is painful
• $\sin(x^2)$ — can't use basic trig rule directly
• $e^{3x}$ — the exponent isn't just $x$

The Chain Rule handles ALL of these.

The Chain Rule Formula

$\boxed{\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)}$

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

Leibniz Notation

If $y = f(u)$ where $u = g(x)$, then:

$\boxed{\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}}$

Key Fact: The Chain Rule is needed whenever you see a function INSIDE another function. It appears in ~80% of all derivative problems on the AP exam.

Worked Examples — Step by Step

Example 1: Find $\frac{d}{dx}(3x+1)^5$

Example 2: Find $\frac{d}{dx}\sin(x^2)$

Example 3: Find $\frac{d}{dx}\sqrt{x^2 + 1}$

Rewrite: $\sqrt{x^2+1} = (x^2+1)^{1/2}$

Example 4: Find $\frac{d}{dx}e^{-x^2}$

AP Tip: The most common Chain Rule error is forgetting to multiply by the inner derivative. Always ask: "Did I multiply by the derivative of what's inside?"

Apply the Chain Rule 🎯

Chain Rule Pattern Reference

Key Concept: Every basic derivative rule has a "chain rule version" where you multiply by the inner derivative.

The "Stuff" Method (Quick Shorthand)

Replace the inner function with "stuff":

AP Tip: This shorthand method is how most students actually think about Chain Rule on the exam. Practice until it's automatic!

More Chain Rule Practice 🎯

Identify the Outer Function 🔍

For each composite function, select the correct outer function.

Chain Rule computation. ✍️

Key Takeaways — Part 1

$\boxed{\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)}$

The #1 Chain Rule mistake: Forgetting the inner derivative. ALWAYS multiply by $g'(x)$!

Up Next: Part 2 — Nested Functions & the Double Chain Rule.

🔗 Nested Functions & Double Chain Rule

Part 2 of 7 — Nested Functions

When the Chain Rule Applies Twice

Some functions have three or more layers. For example:

$f(x) = \sin^2(3x) = [\sin(3x)]^2$

Here we have three layers:

1. Outermost: $u^2$
2. Middle: $\sin(v)$
3. Innermost: $v = 3x$

The general formula for nested compositions:

$\boxed{\frac{d}{dx}[f(g(h(x)))] = f'(g(h(x))) \cdot g'(h(x)) \cdot h'(x)}$

Key Concept: Each layer contributes one factor. Count layers = count factors in the derivative.

Worked Examples — Layer-by-Layer

Example 1: $\frac{d}{dx}[\sin(3x)]^2$

Bonus: By the double-angle identity, $6\sin(3x)\cos(3x) = 3\sin(6x)$.

Example 2: $\frac{d}{dx}e^{\cos(2x)}$

Example 3: $\frac{d}{dx}\sqrt{\ln x} = \frac{d}{dx}(\ln x)^{1/2}$

Example 4: $\frac{d}{dx}\sin^3(2x+1) = \frac{d}{dx}[\sin(2x+1)]^3$

AP Tip: Students commonly write $2\sin(3x)\cos(3x)$ for $\frac{d}{dx}\sin^2(3x)$ — forgetting the factor of $3$ from the innermost layer. Always ask: "Is there another layer inside?"

Nested Chain Rule Practice 🎯

How Many Chain Rule Applications?

Key Fact: The number of chain rule applications = number of layers minus 1.

Strategy: Peel from the Outside In

$\boxed{\text{Step 1: Identify outermost operation. Step 2: Differentiate it. Step 3: Repeat for each inner layer. Step 4: Multiply all factors.}}$

Common Nested Patterns on the AP Exam

Multi-Layer Problems 🎯

How many Chain Rule applications? 🔍

For each function, select how many times you must apply the Chain Rule.

Double Chain Rule computation. ✍️

Key Takeaways — Part 2

$\boxed{\frac{d}{dx}[f(g(h(x)))] = f'(g(h(x))) \cdot g'(h(x)) \cdot h'(x)}$

Up Next: Part 3 — Implicit Differentiation using the Chain Rule.

🔗 Implicit Differentiation

Part 3 of 7 — Implicit Differentiation

What Is Implicit Differentiation?

Sometimes a relationship between $x$ and $y$ is not solved for $y$. For example:

$x^2 + y^2 = 25$

This is a circle. We cannot easily write $y$ as a single function of $x$. But we can still find $\frac{dy}{dx}$ using the Chain Rule.

The Key Idea

$\boxed{\text{Differentiate both sides with respect to } x.\text{ Every time you differentiate } y, \text{ multiply by } \frac{dy}{dx}.}$

Why? Because $y$ is implicitly a function of $x$, so:

$\frac{d}{dx}[y^n] = ny^{n-1} \cdot \frac{dy}{dx}$

Key Concept: Implicit differentiation is just the Chain Rule applied to $y$, treating $y$ as a function of $x$.

Step-by-Step Method

Worked Example 1: Circle

Find $\frac{dy}{dx}$ for $x^2 + y^2 = 25$

Notice: the derivative depends on BOTH $x$ and $y$. This is typical for implicit differentiation.

Worked Example 2: Folium of Descartes

Find $\frac{dy}{dx}$ for $x^3 + y^3 = 6xy$

AP Tip: Implicit differentiation appears frequently on the AP exam, especially when finding slopes of tangent lines to curves.

Practice Implicit Differentiation 🎯

Implicit Differentiation with Trig Functions

Example 3: $\sin(y) = x$

Key Fact: This is exactly how we derive the formula $\frac{d}{dx}[\arcsin(x)] = \frac{1}{\sqrt{1-x^2}}$. Since $\sin(y) = x$, we know $\cos(y) = \sqrt{1-x^2}$.

Tangent Line Applications

Example 4: Find the slope of the tangent line to $x^2 + xy + y^2 = 7$ at $(1, 2)$.

$\boxed{\text{Tangent line: } y - 2 = -\frac{4}{5}(x - 1)}$

Which Rules Are Needed?

Implicit Differentiation Applications 🎯

Second Derivatives (Implicit)

On the AP exam, you may be asked to find $\frac{d^2y}{dx^2}$ implicitly.

Example: For $x^2 + y^2 = 25$, find $\frac{d^2y}{dx^2}$.

We already found $\frac{dy}{dx} = -\frac{x}{y}$. Now differentiate again using quotient rule:

$\frac{d^2y}{dx^2} = \frac{d}{dx}\left(-\frac{x}{y}\right) = -\frac{y(1) - x\frac{dy}{dx}}{y^2}$

Substitute $\frac{dy}{dx} = -\frac{x}{y}$:

$= -\frac{y - x\left(-\frac{x}{y}\right)}{y^2} = -\frac{y + \frac{x^2}{y}}{y^2} = -\frac{\frac{y^2+x^2}{y}}{y^2} = -\frac{x^2+y^2}{y^3}$

Since $x^2+y^2 = 25$:

$\boxed{\frac{d^2y}{dx^2} = -\frac{25}{y^3}}$

AP Tip: When finding $\frac{d^2y}{dx^2}$, substitute the original $\frac{dy}{dx}$ expression AND use the original equation to simplify. This is a common free-response technique.

Which differentiation rule is needed? 🔍

For each term (when differentiating with respect to $x$), select the rule needed.

Implicit Differentiation computation. ✍️

Key Takeaways — Part 3

$\boxed{\text{Every time you differentiate } y, \text{ attach } \frac{dy}{dx}}$

Common Errors:

• Forgetting $\frac{dy}{dx}$ on $y$ terms
• Missing the product rule on $xy$ terms
• Plugging in the point too early (always find general $\frac{dy}{dx}$ first)

Up Next: Part 4 — Related Rates (using implicit differentiation with respect to time).

🔗 Related Rates

Part 4 of 7 — Related Rates

What Are Related Rates?

In related rates problems, two or more quantities are changing with respect to time ($t$), and they are connected by an equation. We use implicit differentiation (with respect to $t$) to find how fast one quantity changes given information about the other.

Key Concept: Related Rates = Implicit Differentiation with respect to time instead of $x$.

The Strategy

$\boxed{\text{Draw} \to \text{Equation} \to \text{Differentiate (w.r.t. } t\text{)} \to \text{Substitute} \to \text{Solve}}$

AP Tip (Critical): NEVER substitute numerical values before differentiating. This is the #1 related rates mistake.

Worked Example 1: Expanding Circle

A stone is dropped in a pond. The circular ripple expands so that its radius increases at $2$ ft/s. How fast is the area increasing when the radius is $5$ ft?

Worked Example 2: Ladder Problem

A 13-ft ladder leans against a wall. The bottom slides away at 2 ft/s. How fast is the top sliding down when the bottom is 5 ft from the wall?

The negative sign means the top is sliding down at $\frac{5}{6}$ ft/s.

Worked Example 3: Conical Tank

Water drains from a conical tank (vertex down) at $2$ ft$^3$/min. The cone has radius 3 ft and height 6 ft. How fast is the water level dropping when the depth is 4 ft?

Similar triangles: $\frac{r}{h} = \frac{3}{6} = \frac{1}{2}$, so $r = \frac{h}{2}$.

Solve These Related Rates Problems 🎯

Essential Geometry Formulas for Related Rates

Common Related Rates Mistakes

Advanced Related Rates 🎯

Related Rates Setup 🔍

Match each scenario with the correct geometric relationship.

Related Rates computation. ✍️

Key Takeaways — Part 4

$\boxed{\text{Related Rates = Implicit Differentiation with respect to } t}$

Remember: Differentiate FIRST, substitute AFTER!

Up Next: Part 5 — Advanced Chain Rule Applications (Logarithmic Differentiation & Inverse Trig).

🔗 Advanced Chain Rule Applications

Part 5 of 7 — Logarithmic Differentiation & Inverse Trig

Logarithmic Differentiation

For functions like $y = x^x$ or $y = (\sin x)^{\cos x}$, standard rules fail because both the base AND exponent depend on $x$. Logarithmic differentiation handles these:

$\boxed{\text{Step 1: } \ln(\text{both sides}) \quad \text{Step 2: Simplify} \quad \text{Step 3: Differentiate implicitly} \quad \text{Step 4: Solve for } \frac{dy}{dx}}$

Key Concept: When do you need log differentiation?

• Variable base AND variable exponent: $f(x)^{g(x)}$
• Products/quotients of many factors (to simplify)

When to Use Each Technique

Worked Example 1: $\frac{d}{dx}x^x$

$\boxed{\frac{d}{dx}x^x = x^x(\ln x + 1)}$

Worked Example 2: Simplifying Complex Products

Find $\frac{d}{dx}\frac{x^2\sqrt{x+1}}{(2x-3)^4}$

AP Tip: Log differentiation is rarely tested directly on AP Calc AB, but it's an important tool for AP Calc BC and is excellent for building understanding.

Logarithmic Differentiation 🎯

Chain Rule with Inverse Trig Functions

The inverse trig derivatives all require the Chain Rule when the argument is a composite:

Key Fact: On the AP exam, $\arctan$ is the most commonly tested inverse trig function. Know its derivative cold.

Worked Examples

Example 3: $\frac{d}{dx}\arctan(3x) = \frac{3}{1+(3x)^2} = \frac{3}{1+9x^2}$

Example 4: $\frac{d}{dx}\arcsin(x^2) = \frac{2x}{\sqrt{1-(x^2)^2}} = \frac{2x}{\sqrt{1-x^4}}$

Example 5: $\frac{d}{dx}\arctan(e^x) = \frac{e^x}{1+(e^x)^2} = \frac{e^x}{1+e^{2x}}$

Inverse Trig Derivatives 🎯

Match the derivative technique 🔍

For each function, select the best approach.

Inverse trig computation. ✍️

Key Takeaways — Part 5

$\boxed{\frac{d}{dx}[\arctan(u)] = \frac{u'}{1+u^2} \qquad \frac{d}{dx}[\arcsin(u)] = \frac{u'}{\sqrt{1-u^2}}}$

Up Next: Part 6 — Problem-Solving Workshop with mixed Chain Rule problems.

🔗 Problem-Solving Workshop

Part 6 of 7 — Mixed Chain Rule Problems

Decision Framework

Every derivative problem begins with the same question: What is the outermost operation?

$\boxed{\text{Identify outermost operation} \to \text{Apply rule} \to \text{Chain Rule for inner layers}}$

Key Strategy: Work from the OUTSIDE IN. The outermost operation determines which rule to start with.

Combining Chain Rule with Other Rules

Example 1: Product + Chain

$\frac{d}{dx}[x^2 \sin(3x)] = 2x\sin(3x) + x^2 \cdot \cos(3x) \cdot 3 = 2x\sin(3x) + 3x^2\cos(3x)$

Example 2: Quotient + Chain

$\frac{d}{dx}\left[\frac{e^{2x}}{x+1}\right] = \frac{2e^{2x}(x+1) - e^{2x}}{(x+1)^2} = \frac{e^{2x}(2x+1)}{(x+1)^2}$

Example 3: Chain on $\ln$

$\frac{d}{dx}[\ln(\cos x)] = \frac{-\sin x}{\cos x} = -\tan x$

Important: $\frac{d}{dx}\frac{\sin x}{e^x}$ (Shortcut)

$\frac{d}{dx}\left[\frac{\sin x}{e^x}\right] = \frac{d}{dx}[\sin x \cdot e^{-x}] = \cos x \cdot e^{-x} + \sin x \cdot (-e^{-x}) = \frac{\cos x - \sin x}{e^x}$

AP Tip: Sometimes rewriting $\frac{f(x)}{e^x}$ as $f(x) \cdot e^{-x}$ and using Product Rule is easier than Quotient Rule.

AP-Style Problems — Set 1 🎯

Table-Based Chain Rule Problems

On the AP exam, you may be given a table of values and asked to compute a composite function's derivative.

Key Fact: If $h(x) = f(g(x))$, then $h'(a) = f'(g(a)) \cdot g'(a)$. You need: $g(a)$ from the table, then $f'$ at that value, then $g'(a)$.

Example: Find $h'(1)$ where $h(x) = f(g(x))$.

$h'(1) = f'(g(1)) \cdot g'(1) = f'(2) \cdot 4 = 1 \cdot 4 = 4$

Example: Find $k'(2)$ where $k(x) = g(f(x))$.

$k'(2) = g'(f(2)) \cdot f'(2) = g'(5) \cdot 1$

But $g'(5)$ is not in the table — insufficient information!

AP Tip: Always check that the required values are in the table before computing. If $g(a)$ gives a value not in the table, you can't find $f'(g(a))$.

Table-Based & Mixed Problems 🎯

Use the table from the previous section.

Subtle Distinctions

These look similar but have very different derivatives:

Key Principle: Know the difference between: constant exponent (Power Rule), constant base (Exponential Rule), both variable (Log Differentiation).

FTC Part 1 + Chain Rule

$\boxed{\frac{d}{dx}\int_a^{g(x)} f(t)\,dt = f(g(x)) \cdot g'(x)}$

Example: $\frac{d}{dx}\int_0^{x^2} \sin(t)\,dt = \sin(x^2) \cdot 2x = 2x\sin(x^2)$

Identify the correct derivative 🔍

Match each function to its derivative.

Table-based Chain Rule computation. ✍️

Workshop Complete!

Up Next: Part 7 — Comprehensive Review & Final Assessment.

🔗 Chain Rule — Comprehensive Review

Part 7 of 7 — Review & Final Assessment

Complete Chain Rule Summary

$\boxed{\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)}$

AP Exam Frequency

Key Fact: The Chain Rule appears in ~80% of all derivative problems on the AP exam. It is embedded in:

• All implicit differentiation problems
• All related rates problems
• FTC Part 1 with variable upper limit
• Most trig, exponential, and logarithmic derivatives

Quick Reference — All Chain Rule Patterns

Common Errors to Avoid