Generalized Power Rule

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The Most Common Chain Rule Pattern

You already know the Power Rule: $\frac{d}{dx}(x^n) = nx^{n-1}$. But what about $(x^2 + 1)^{10}$? You cannot just write $10(x^2+1)^9$. That ignores what is inside the parentheses.

The Generalized Power Rule fixes this by combining the Power Rule with the Chain Rule. It's arguably the most frequently used differentiation formula after the basic rules.

$$\frac{d}{dx}[g(x)]^n = n[g(x)]^{n-1} \cdot g'(x)$$

Key insight: Bring down the exponent, reduce by one, then multiply by the derivative of what's inside.

Prerequisite Map

Quick Reference

Property	Value
Concept	Chain Rule
Chapter	2.5
Difficulty	Intermediate
Time	~15 minutes

Key Concepts

The Formula

For any differentiable function $g(x)$ and any real number $n$:

$$\boxed{\frac{d}{dx}[g(x)]^n = n[g(x)]^{n-1} \cdot g'(x)}$$

Why This Works

This is just the Chain Rule where the outer function is $f(u) = u^n$:

Component	Value	Derivative
Outer: $f(u)$	$u^n$	$nu^{n-1}$
Inner: $g(x)$	whatever's inside	$g'(x)$

Chain Rule: $\frac{d}{dx}[g(x)]^n = f'(g(x)) \cdot g'(x) = n[g(x)]^{n-1} \cdot g'(x)$

The Three-Step Process

For any expression of the form $(\text{stuff})^n$:

1. Bring down the exponent as a coefficient
2. Reduce the exponent by 1, keeping the inside unchanged
3. Multiply by the derivative of the inside

Example: $\frac{d}{dx}(x^3 + 2x)^7$

Step 1: Bring down 7         →  7
Step 2: Reduce exponent      →  7(x³ + 2x)⁶
Step 3: Multiply by inside'  →  7(x³ + 2x)⁶ · (3x² + 2)

Special Cases with Fractional and Negative Exponents

The formula works for all real exponents:

Expression	Rewrite	Derivative
$\sqrt{g(x)}$	$[g(x)]^{1/2}$	$\frac{1}{2}[g(x)]^{-1/2} \cdot g'(x) = \frac{g'(x)}{2\sqrt{g(x)}}$
$\sqrt[3]{g(x)}$	$[g(x)]^{1/3}$	$\frac{1}{3}[g(x)]^{-2/3} \cdot g'(x)$
$\frac{1}{g(x)}$	$[g(x)]^{-1}$	$-[g(x)]^{-2} \cdot g'(x) = \frac{-g'(x)}{[g(x)]^2}$
$\frac{1}{[g(x)]^2}$	$[g(x)]^{-2}$	$-2[g(x)]^{-3} \cdot g'(x)$

The Common Error

Wrong: $\frac{d}{dx}(x^2 + 1)^5 = 5(x^2+1)^4$ ✗

This forgets the Chain Rule! The derivative of $x^2 + 1$ is $2x$, not 1.

Correct: $\frac{d}{dx}(x^2 + 1)^5 = 5(x^2+1)^4 \cdot 2x = 10x(x^2+1)^4$ ✓

Rule of thumb: If there's anything other than just "$x$" inside the parentheses, you need to multiply by its derivative.

Practice Problems

CCI-Style Conceptual Questions

Mastery Checklist

• [ ] I can apply the Generalized Power Rule to expressions like $(2x+3)^5$
• [ ] I can handle fractional exponents (square roots, cube roots)
• [ ] I can handle negative exponents (reciprocals)
• [ ] I remember to multiply by the derivative of the inside
• [ ] I can combine this rule with Product or Quotient Rules

Mental Model

The Gift-Wrapping Analogy: Think of $[g(x)]^n$ as a gift inside $n$ layers of wrapping paper.

1. Unwrap one layer (bring down $n$, reduce exponent by 1)
2. Account for the gift inside (multiply by the derivative of what's wrapped)

If you forget step 2, you're only describing the wrapping paper, not what's actually changing inside!

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Connections

Looking back:

• Chain Rule Formula is the general principle; this is its most common application

Looking ahead:

• Implicit Differentiation applies this to expressions like $y^3$ where $y$ is a function of $x$
• Related Rates problems often involve powers of related quantities

Why this pattern is everywhere: Powers appear constantly in applications: areas ($r^2$), volumes ($r^3$), inverse-square laws ($1/r^2$), and growth models. Every time the base is a function, you need this rule.

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Chain Rule Formula	Skills Index	Implicit Differentiation

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Last updated: 2026-01-22