Chain Rule for MATH 122
Exam Relevance for MATH 122
A core exam topic. Expect 2-3 questions requiring chain rule, often combined with exponential/log derivatives.
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What is the Chain Rule?
The chain rule is how you differentiate composite functions — functions inside other functions.
Example of a composite function: $f(x) = (3x + 1)^5$
Here, there's an "outer function" (raising to the 5th power) and an "inner function" ($3x + 1$).
The problem: You can't just use the power rule directly. $(3x+1)^5 \neq 5(3x+1)^4$ — that's missing something!
The Formula
If $y = f(g(x))$, then:
$$\frac{dy}{dx} = f'(g(x)) \cdot g'(x)$$
In words: "Derivative of the outer (with the inner unchanged) times derivative of the inner."
Alternative notation (Leibniz): If $y = f(u)$ and $u = g(x)$:
$$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$$
How to Identify When You Need the Chain Rule
Ask yourself: "Is there a function inside another function?"
| Expression | Outer Function | Inner Function | Need Chain Rule? |
|---|---|---|---|
| $(2x+1)^3$ | $(\cdot)^3$ | $2x+1$ | ✅ Yes |
| $\sin(x^2)$ | $\sin(\cdot)$ | $x^2$ | ✅ Yes |
| $e^{5x}$ | $e^{(\cdot)}$ | $5x$ | ✅ Yes |
| $x^3$ | — | — | ❌ No (just power rule) |
| $\sin(x)$ | — | — | ❌ No (inner is just $x$) |
Problem: Find $\frac{d}{dx}[(3x+1)^5]$
Step 1: Identify outer and inner functions:
- Outer: $f(u) = u^5$
- Inner: $u = g(x) = 3x + 1$
Step 2: Find each derivative:
- $f'(u) = 5u^4$
- $g'(x) = 3$
Step 3: Apply chain rule — outer derivative (keep inner unchanged) times inner derivative: $$\frac{d}{dx}[(3x+1)^5] = 5(3x+1)^4 \cdot 3$$
$$\boxed{15(3x+1)^4}$$
Problem: Find $\frac{d}{dx}[(x^2 - 4x)^3]$
Step 1: Identify the parts:
- Outer: $u^3$
- Inner: $u = x^2 - 4x$
Step 2: Find derivatives:
- Outer derivative: $3u^2$
- Inner derivative: $2x - 4$
Step 3: Apply chain rule: $$\frac{d}{dx}[(x^2-4x)^3] = 3(x^2-4x)^2 \cdot (2x-4)$$
$$\boxed{3(x^2-4x)^2(2x-4)}$$
Or factored: $6(x^2-4x)^2(x-2)$
Problem: Find $\frac{d}{dx}[\sin(x^2)]$
Step 1: Identify the parts:
- Outer: $\sin(u)$
- Inner: $u = x^2$
Step 2: Find derivatives:
- Outer derivative: $\cos(u)$
- Inner derivative: $2x$
Step 3: Apply chain rule: $$\frac{d}{dx}[\sin(x^2)] = \cos(x^2) \cdot 2x$$
$$\boxed{2x\cos(x^2)}$$
Problem: Find $\frac{d}{dx}[e^{3x}]$
Step 1: Identify the parts:
- Outer: $e^u$
- Inner: $u = 3x$
Step 2: Find derivatives:
- Outer derivative: $e^u$ (exponential is its own derivative!)
- Inner derivative: $3$
Step 3: Apply chain rule: $$\frac{d}{dx}[e^{3x}] = e^{3x} \cdot 3$$
$$\boxed{3e^{3x}}$$
Problem: Find $\frac{d}{dx}[\sqrt{x^2+1}]$
Step 1: Rewrite as a power: $$\sqrt{x^2+1} = (x^2+1)^{1/2}$$
Step 2: Identify the parts:
- Outer: $u^{1/2}$
- Inner: $u = x^2 + 1$
Step 3: Find derivatives:
- Outer derivative: $\frac{1}{2}u^{-1/2} = \frac{1}{2\sqrt{u}}$
- Inner derivative: $2x$
Step 4: Apply chain rule: $$\frac{d}{dx}[\sqrt{x^2+1}] = \frac{1}{2\sqrt{x^2+1}} \cdot 2x = \frac{2x}{2\sqrt{x^2+1}}$$
$$\boxed{\frac{x}{\sqrt{x^2+1}}}$$
Problem: Find $\frac{d}{dx}[\sin^2(3x)]$
Step 1: Recognize the structure — this is $[\sin(3x)]^2$, which has THREE layers:
- Outermost: $(\cdot)^2$
- Middle: $\sin(\cdot)$
- Innermost: $3x$
Step 2: Work from outside in:
First, derivative of outer $u^2$ is $2u$, where $u = \sin(3x)$: $$2\sin(3x) \cdot \frac{d}{dx}[\sin(3x)]$$
Then, $\frac{d}{dx}[\sin(3x)] = \cos(3x) \cdot 3$
Step 3: Combine: $$\frac{d}{dx}[\sin^2(3x)] = 2\sin(3x) \cdot \cos(3x) \cdot 3$$
$$\boxed{6\sin(3x)\cos(3x)}$$
Or using the double angle identity: $\boxed{3\sin(6x)}$
Quick Reference: Common Chain Rule Patterns
| Function | Derivative |
|---|---|
| $[f(x)]^n$ | $n[f(x)]^{n-1} \cdot f'(x)$ |
| $\sin(f(x))$ | $\cos(f(x)) \cdot f'(x)$ |
| $\cos(f(x))$ | $-\sin(f(x)) \cdot f'(x)$ |
| $e^{f(x)}$ | $e^{f(x)} \cdot f'(x)$ |
| $\ln(f(x))$ | $\frac{f'(x)}{f(x)}$ |
| $\sqrt{f(x)}$ | $\frac{f'(x)}{2\sqrt{f(x)}}$ |
Common Mistakes and Misunderstandings
❌ Mistake: Forgetting the chain rule entirely
Wrong: $\frac{d}{dx}[(2x+1)^4] = 4(2x+1)^3$
Why it's wrong: You used the power rule on the outer function but forgot to multiply by the derivative of the inner function ($2x+1$).
Correct: $\frac{d}{dx}[(2x+1)^4] = 4(2x+1)^3 \cdot 2 = 8(2x+1)^3$
❌ Mistake: Changing the inner function when differentiating
Wrong: $\frac{d}{dx}[\sin(x^2)] = \cos(2x) \cdot 2x$
Why it's wrong: The inner function $x^2$ should stay as $x^2$ inside the cosine, not become $2x$.
Correct: $\frac{d}{dx}[\sin(x^2)] = \cos(x^2) \cdot 2x$
❌ Mistake: Applying chain rule when it's not needed
Wrong: $\frac{d}{dx}[x^3] = 3x^2 \cdot 1 = 3x^2$ (overthinking it)
Why it's unnecessary: When the inner function is just $x$, its derivative is 1, so the chain rule reduces to the basic rule. You don't need to explicitly write "$\cdot 1$".
Simpler: $\frac{d}{dx}[x^3] = 3x^2$ (just use the power rule directly)
Chain Rule (Standard Form)
Derivative of a composite function: take the derivative of the outer function (keeping the inner function unchanged), then multiply by the derivative of the inner function
Variables:
- $f$:
- outer function
- $g(x)$:
- inner function
- $f'(g(x))$:
- derivative of outer, evaluated at the inner
- $g'(x)$:
- derivative of the inner function
Chain Rule (Leibniz Notation)
If y depends on u and u depends on x, the derivative of y with respect to x is the product of the two intermediate derivatives
Variables:
- $y$:
- dependent variable (function of u)
- $u$:
- intermediate variable (function of x)
- $x$:
- independent variable
Generalized Power Rule
Chain rule applied to a function raised to a power — most common use case!
Variables:
- $f(x)$:
- the inner function (base)
- $n$:
- the exponent (any real number)
- $f'(x)$:
- derivative of the inner function
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