Basic Differentiation Rules

📐 Basic Differentiation Rules

Part 1 of 7 — The Power Rule & Foundational Rules

Welcome to differentiation! This topic covers the rules you'll use on every single calculus problem.

The Power Rule

The single most important differentiation rule:

$\boxed{\frac{d}{dx}[x^n] = n x^{n-1} \quad \text{for any real } n}$

Key Fact: The Power Rule works for ALL real exponents — positive, negative, fractional, irrational, even $n = 0$. When $n = 0$: $\frac{d}{dx}[x^0] = \frac{d}{dx}[1] = 0$.

Step-by-Step Process

1. Identify the exponent $n$
2. Bring $n$ down as a coefficient (multiply)
3. Subtract 1 from the exponent

Constant Rule & Constant Multiple Rule

$\boxed{\frac{d}{dx}[c] = 0 \qquad \frac{d}{dx}[c \cdot f(x)] = c \cdot f'(x)}$

Constants vanish when differentiated. Constants attached to functions "pass through" the derivative.

⚠️ Common Mistake: Students sometimes think $\frac{d}{dx}[\pi^2] = 2\pi$. Remember: $\pi$ is a constant, not a variable! The derivative is $0$.

Sum & Difference Rule

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

Differentiate each term independently — linearity of the derivative.

Worked Example

Find $\frac{d}{dx}(3x^4 - 5x^2 + 7x - 2)$

$\frac{d}{dx}(3x^4 - 5x^2 + 7x - 2) = 12x^3 - 10x + 7$

Apply the Power Rule 🎯

Negative and Fractional Exponents

Key Principle: Before applying the Power Rule, rewrite all roots, fractions, and radicals using exponential notation.

Rewrite Rules:

$\frac{1}{x^n} = x^{-n} \qquad \sqrt[n]{x^m} = x^{m/n}$

Worked Example

Find $\frac{d}{dx}\left(\frac{3}{x^2} + 4\sqrt{x} - \frac{7}{\sqrt[3]{x}}\right)$

Step 1 — Rewrite: $3x^{-2} + 4x^{1/2} - 7x^{-1/3}$

Step 2 — Differentiate: $-6x^{-3} + 2x^{-1/2} + \frac{7}{3}x^{-4/3}$

Step 3 — Simplify: $-\frac{6}{x^3} + \frac{2}{\sqrt{x}} + \frac{7}{3\sqrt[3]{x^4}}$

AP Tip: On the AP exam, you do NOT need to simplify your answer. Leaving the derivative in negative-exponent form is perfectly acceptable and saves time!

Negative & Fractional Exponents 🎯

Special Derivatives to Memorize

Beyond the Power Rule, these constants arise frequently:

$\boxed{\frac{d}{dx}[e^x] = e^x \qquad \frac{d}{dx}[\ln x] = \frac{1}{x}}$

$\boxed{\frac{d}{dx}[a^x] = a^x \ln a \qquad \frac{d}{dx}[\log_a x] = \frac{1}{x \ln a}}$

Key Fact: $e^x$ is the only function that equals its own derivative (up to constant multiples). This is why $e$ is so special in calculus!

Match each function to its derivative.

Find the derivative and evaluate. ✍️

Key Takeaways — Part 1

Workflow for any polynomial/power derivative:

1. Rewrite all roots and fractions as power expressions
2. Apply Power Rule term by term (bring down exponent, subtract 1)
3. Simplify if desired (not required on AP exam)

Up Next: Part 2 — The Product Rule, for when two functions are multiplied together.

📐 The Product Rule

Part 2 of 7 — Product Rule

Why Can't We Just Multiply the Derivatives?

A common (and dangerous) mistake:

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

Quick proof it fails: $\frac{d}{dx}[x \cdot x] = \frac{d}{dx}[x^2] = 2x$, but $1 \cdot 1 = 1 \neq 2x$. ✗

The Product Rule

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

Memory aids:

• "Derivative of first times second, plus first times derivative of second"
• Short form: $(fg)' = f'g + fg'$
• Leibniz form: $d(uv) = u\,dv + v\,du$

Key Fact: The Product Rule comes from the limit definition. The "extra" term $fg'$ accounts for the fact that both factors are changing simultaneously.

Worked Examples — Product Rule

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

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

Example 2: Find $\frac{d}{dx}[e^x \ln x]$

$= e^x\left(\ln x + \frac{1}{x}\right)$

AP Tip: Factor common terms in your final answer when possible. Graders appreciate clean answers, and factoring helps with sign analysis later.

Example 3: Find $\frac{d}{dx}[xe^x]$ — the most commonly tested product!

$\frac{d}{dx}[xe^x] = (1)e^x + x(e^x) = e^x(1 + x)$

At $x = 0$: $f'(0) = e^0(1+0) = 1$. At $x = -1$: $f'(-1) = e^{-1}(0) = 0$ (this is a critical point!)

Apply the Product Rule 🎯

When to Use Product Rule vs. Expand First

Examples:

Product Rule with Tables (AP Exam Favorite!)

If $f(2) = 3$, $f'(2) = -1$, $g(2) = 4$, $g'(2) = 5$, find $\frac{d}{dx}[f(x)g(x)]$ at $x = 2$:

$f'(2)g(2) + f(2)g'(2) = (-1)(4) + (3)(5) = -4 + 15 = 11$

AP Tip: Table-based derivative problems appear on almost EVERY AP exam. Practice reading values from tables and plugging into Product Rule.

Product Rule Challenge 🎯

Extended Product Rule — Three or More Factors

For three functions:

$\frac{d}{dx}[f \cdot g \cdot h] = f'gh + fg'h + fgh'$

Pattern: Each factor takes a turn being differentiated while the others stay.

Example: Find $\frac{d}{dx}[x^2 \sin x \cdot e^x]$

$= 2x \sin x \cdot e^x + x^2 \cos x \cdot e^x + x^2 \sin x \cdot e^x$

$= e^x(2x\sin x + x^2 \cos x + x^2 \sin x)$

Key Concept: You can also apply the two-factor Product Rule twice: treat $(fg)$ as one function and apply Product Rule with $h$. This gives the same result.

Product Rule with given values.

Product Rule computation. ✍️

Key Takeaways — Part 2

Up Next: Part 3 — The Quotient Rule, for derivatives of fractions.

📐 The Quotient Rule

Part 3 of 7 — Quotient Rule

The Quotient Rule

$\boxed{\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)\,g(x) - f(x)\,g'(x)}{[g(x)]^2}}$

Memory aids:

• "Low d-High minus High d-Low, all over Low squared"
• Short form: $\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}$

⚠️ Critical Warning: The minus sign in the numerator is the #1 source of errors. The order matters — it's $f'g$ MINUS $fg'$, not the other way around. Unlike the Product Rule, the Quotient Rule is NOT symmetric!

Comparison: Product Rule vs. Quotient Rule

Worked Examples

Example 1: Find $\frac{d}{dx}\frac{x^2}{\sin x}$

$\frac{d}{dx}\frac{x^2}{\sin x} = \frac{2x\sin x - x^2\cos x}{\sin^2 x}$

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

$= \frac{e^x(x+1) - e^x(1)}{(x+1)^2} = \frac{e^x(x+1-1)}{(x+1)^2} = \frac{xe^x}{(x+1)^2}$

AP Tip: Always look for common factors in the numerator after applying the Quotient Rule. Simplifying makes it easier to find critical points and sign analysis.

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

$\frac{1 \cdot (x+2) - x \cdot 1}{(x+2)^2} = \frac{x + 2 - x}{(x+2)^2} = \frac{2}{(x+2)^2}$

Key Fact: When $\frac{d}{dx}\frac{x}{x+c}$ yields a positive constant over a square, the function is always increasing. This is useful for sign analysis!

Apply the Quotient Rule 🎯

When to Avoid the Quotient Rule

The Quotient Rule is powerful but often overkill. Use smarter alternatives when possible:

Splitting Fractions — A Powerful Technique

$\frac{x^3 + 6x^2 - 2}{x^2} = x + 6 - 2x^{-2}$

Now differentiate term by term: $1 + 0 + 4x^{-3} = 1 + \frac{4}{x^3}$

Compare to using Quotient Rule on the original — much more work for the same answer!

Deriving Trig Derivatives via Quotient Rule

$\frac{d}{dx}[\tan x] = \frac{d}{dx}\frac{\sin x}{\cos x} = \frac{\cos x \cdot \cos x - \sin x(-\sin x)}{\cos^2 x} = \frac{\cos^2 x + \sin^2 x}{\cos^2 x} = \frac{1}{\cos^2 x} = \sec^2 x$

Key Concept: The Quotient Rule is how we derive the derivatives of $\tan x$, $\cot x$, $\sec x$, and $\csc x$ from $\sin x$ and $\cos x$.

Quotient Rule Mastery 🎯

Quotient Rule with Tables (AP Exam Staple)

Given:

Find $\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right]$ at $x = 1$:

$\frac{f'(1)g(1) - f(1)g'(1)}{[g(1)]^2} = \frac{(-2)(4) - (3)(5)}{4^2} = \frac{-8 - 15}{16} = \frac{-23}{16}$

Find $\frac{d}{dx}\left[\frac{g(x)}{f(x)}\right]$ at $x = 2$:

$\frac{g'(2)f(2) - g(2)f'(2)}{[f(2)]^2} = \frac{(-3)(-1) - (2)(6)}{(-1)^2} = \frac{3 - 12}{1} = -9$

AP Tip: Watch for problems that ask for $\frac{d}{dx}\left[\frac{g}{f}\right]$ instead of $\frac{d}{dx}\left[\frac{f}{g}\right]$ — swapping the roles of $f$ and $g$ is a common trap!

Choose the best differentiation strategy.

Quotient Rule computation. ✍️

Key Takeaways — Part 3

Decision Tree: Which Rule?

Up Next: Part 4 — Trigonometric Derivatives in depth.

📐 Trigonometric Derivatives

Part 4 of 7 — Trig Derivatives

The Six Trigonometric Derivatives

These must be memorized perfectly for the AP exam:

$\boxed{\frac{d}{dx}[\sin x] = \cos x \qquad \frac{d}{dx}[\cos x] = -\sin x}$

$\boxed{\frac{d}{dx}[\tan x] = \sec^2 x \qquad \frac{d}{dx}[\cot x] = -\csc^2 x}$

$\boxed{\frac{d}{dx}[\sec x] = \sec x \tan x \qquad \frac{d}{dx}[\csc x] = -\csc x \cot x}$

Pattern Recognition — The Negative Sign Rule

Key Fact: The co-functions (cos, cot, csc) ALL have negative derivatives. The regular functions (sin, tan, sec) have positive derivatives.

Another Pattern — Squared vs. Product

Worked Examples — Basic Trig Derivatives

Key Angle Values Reference

AP Tip: You need instant recall of trig values at these angles. The derivative questions almost always evaluate at one of these special angles.

Trig Derivatives 🎯

Combining Trig Derivatives with Product & Quotient Rules

Example 1: $\frac{d}{dx}(e^x \sin x)$

Product Rule: $e^x \sin x + e^x \cos x = e^x(\sin x + \cos x)$

At $x = 0$: $e^0(0 + 1) = 1$

Example 2: $\frac{d}{dx}\left(\frac{\tan x}{x}\right)$

Quotient Rule: $\frac{\sec^2 x \cdot x - \tan x \cdot 1}{x^2} = \frac{x\sec^2 x - \tan x}{x^2}$

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

Product Rule: $2x \sec x + x^2 \sec x \tan x = x\sec x(2 + x\tan x)$

Key Concept: When combining trig derivatives with Product/Quotient Rule, always set up the table ($f, f', g, g'$) to stay organized and avoid sign errors.

Mixed Trig Problems 🎯

Where Do Trig Derivatives Come From?

The derivatives of $\sin x$ and $\cos x$ come from the limit definition:

$\frac{d}{dx}[\sin x] = \lim_{h \to 0} \frac{\sin(x+h) - \sin x}{h}$

Using the angle addition formula $\sin(x+h) = \sin x \cos h + \cos x \sin h$:

$= \lim_{h \to 0} \frac{\sin x(\cos h - 1) + \cos x \sin h}{h} = \sin x \cdot 0 + \cos x \cdot 1 = \cos x$

This relies on the special limits: $\lim_{h \to 0}\frac{\sin h}{h} = 1$ and $\lim_{h \to 0}\frac{\cos h - 1}{h} = 0$.

The other four come from $\sin x$ and $\cos x$:

Complete the derivative.

Trig derivative evaluation. ✍️

Key Takeaways — Part 4

Memory checklist:

1. Co-functions → negative sign (cos, cot, csc)
2. tan and cot → squared results ($\sec^2$, $\csc^2$)
3. sec and csc → product results (sec·tan, csc·cot)
4. Know exact trig values at $0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2}$

Up Next: Part 5 — Higher-Order Derivatives.

📐 Higher-Order Derivatives

Part 5 of 7 — Higher-Order Derivatives

What Are Higher-Order Derivatives?

The second derivative is the derivative of the derivative:

$\boxed{f''(x) = \frac{d^2 y}{dx^2} = \frac{d}{dx}\left[\frac{dy}{dx}\right]}$

Notation Comparison

Key Fact: For $n \geq 4$, we write $f^{(n)}(x)$ with parentheses to avoid confusion with powers: $f^{(4)}(x)$ is the 4th derivative, not $[f(x)]^4$.

Physical Interpretation — Motion

Worked Examples

Example 1: Find all derivatives of $f(x) = x^5 - 3x^3 + 2x$

Key Principle: Any polynomial of degree $n$ has $f^{(n+1)}(x) = 0$. The $n$th derivative of $x^n$ is $n!$ (n factorial).

Example 2: Higher derivatives of $e^{2x}$

$\frac{d^n}{dx^n}[e^{2x}] = 2^n e^{2x}$

Each derivative multiplies by 2 (Chain Rule): $y' = 2e^{2x}$, $y'' = 4e^{2x}$, $y''' = 8e^{2x}$, ...

Example 3: The Trig Cycle

$\boxed{\frac{d^n}{dx^n}[\sin x] = \sin\left(x + \frac{n\pi}{2}\right) \qquad \frac{d^n}{dx^n}[\cos x] = \cos\left(x + \frac{n\pi}{2}\right)}$

Find Higher-Order Derivatives 🎯

Concavity and the Second Derivative

The second derivative provides crucial information about the shape of a graph:

⚠️ Critical Warning: $f''(c) = 0$ does NOT guarantee an inflection point! You must verify that $f''$ actually changes sign at $c$. Example: $f(x) = x^4$ has $f''(0) = 0$ but NO inflection point (concave up on both sides).

The Second Derivative Test

At a critical point where $f'(c) = 0$:

$\boxed{f'(c) = 0 \text{ and } f''(c) > 0 \implies \text{local minimum at } x = c}$

Worked Example

Find where $f(x) = x^3 - 3x$ is concave up.

$f''(x) = 6x$.

Concave up when $f''(x) > 0$: $6x > 0 \implies x > 0$.

So $f$ is concave up on $(0, \infty)$ and concave down on $(-\infty, 0)$ with an inflection point at $x = 0$.

Second Derivative Applications 🎯

Connecting $f$, $f'$, and $f''$ — The Big Picture

AP Tip: The AP exam frequently gives you a graph of $f'(x)$ and asks about $f(x)$ or $f''(x)$. Remember: the derivative of $f'$ IS $f''$, so where $f'$ is increasing, $f'' > 0$ (concave up for $f$).

Analyze concavity and extrema.

Higher-order derivative computation. ✍️

Key Takeaways — Part 5