📐 Basic Differentiation Rules

Part 1 of 7 — The Power Rule

The Power Rule

The most fundamental differentiation rule:

$\frac{d}{dx}[x^n] = nx^{n-1}$

This works for any real exponent $n$ — positive, negative, fractional, or zero.

Examples with Positive Integer Exponents

Constant Multiple Rule

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

Constants just "come along for the ride."

Sum/Difference Rule

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

Differentiate term by term.

Worked Example

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

$= 12x^3 - 10x + 7$

Each term differentiated independently: $3(4x^3) - 5(2x) + 7(1) - 0$.

Differentiate using the Power Rule 🎯

Negative and Fractional Exponents

Rewrite first, then apply the Power Rule:

Worked Example

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

$= \frac{d}{dx}(3x^{-2} + 4x^{1/2}) = -6x^{-3} + 2x^{-1/2} = -\frac{6}{x^3} + \frac{2}{\sqrt{x}}$

Negative & Fractional Exponents 🎯

Key Takeaways — Part 1

1. Power Rule: $\frac{d}{dx}x^n = nx^{n-1}$ for any real $n$
2. Constants vanish: $\frac{d}{dx}[c] = 0$
3. Constant multiples pass through: $\frac{d}{dx}[cf(x)] = cf'(x)$
4. Sum/Difference: differentiate term by term
5. Always rewrite roots and fractions as power expressions first

📐 The Product Rule

Part 2 of 7 — Product Rule

Why Can't We Just Multiply the Derivatives?

A common 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

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

Memory aid: "first times derivative of second, plus second times derivative of first" — or simply "$f'g + fg'$".

Worked Example 1

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

Worked Example 2

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

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

Apply the Product Rule 🎯

When to Use Product Rule vs. Expand

Sometimes it is easier to expand first:

• $(2x+1)(x^2-3)$ → expand to $2x^3 + x^2 - 6x - 3$, then differentiate term by term

But when expansion is impractical (e.g., $x^5 e^x$ or $\sin x \ln x$), the Product Rule is essential.

Worked Example 3

Find $\frac{d}{dx}[x e^x]$ and evaluate at $x = 0$

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

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

AP Tip: The Product Rule is frequently tested. Know it cold.

Product Rule Challenge 🎯

Key Takeaways — Part 2

1. Product Rule: $(fg)' = f'g + fg'$
2. Do NOT multiply derivatives: $(fg)' \neq f'g'$
3. Consider expanding if both factors are polynomials
4. Factor common terms in your answer when possible

📐 The Quotient Rule

Part 3 of 7 — Quotient Rule

The Quotient Rule

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

Memory aid: "Low d-High minus High d-Low, over Low squared" — $\frac{gf' - fg'}{g^2}$... actually the standard is $\frac{f'g - fg'}{g^2}$.

Worked Example 1

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

Worked Example 2

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

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

Pro tip: Sometimes you can avoid the Quotient Rule by rewriting: $\frac{1}{x^3} = x^{-3}$, then use the Power Rule.

Apply the Quotient Rule 🎯

Deriving Trig Derivatives via Quotient Rule

The Quotient Rule lets us derive the derivatives of $\tan x$, $\cot x$, $\sec x$, and $\csc x$:

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

When to Avoid the Quotient Rule

If the denominator is just a constant, do NOT use the Quotient Rule: $\frac{d}{dx}\frac{x^3 + 2x}{5} = \frac{1}{5}(3x^2 + 2)$

If you can rewrite as a negative exponent, that is often simpler: $\frac{3}{x^4} = 3x^{-4} \implies \frac{d}{dx} = -12x^{-5}$

Quotient Rule Mastery 🎯

Key Takeaways — Part 3

1. Quotient Rule: $\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}$
2. The minus sign in the numerator is the most common source of errors
3. Avoid the Quotient Rule when the denominator is a constant or a simple power of $x$
4. The Quotient Rule derives all reciprocal trig derivatives

📐 Trigonometric Derivatives

Part 4 of 7 — Trig Derivatives

The Six Trig Derivatives

Pattern Recognition

Notice the negative signs always appear with co-functions (cos, cot, csc).

Worked Examples

Trig Derivatives 🎯

Special Values to Know

At $x = 0$:

• $\sin(0) = 0$, $\cos(0) = 1$
• $\frac{d}{dx}\sin x \big|_{x=0} = \cos(0) = 1$
• $\frac{d}{dx}\cos x \big|_{x=0} = -\sin(0) = 0$

Combining Rules

Find $\frac{d}{dx}(e^x \sin x)$

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

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

Chain Rule: $2\sin x \cdot \cos x = \sin(2x)$

Mixed Trig Problems 🎯

Key Takeaways — Part 4

1. Memorize all six trig derivatives
2. Negatives go with co-functions: cos, cot, csc
3. When trig functions are combined with other functions, use Product/Quotient/Chain rules as needed
4. Know your trig values at key angles: $0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2}$

📐 Higher-Order Derivatives

Part 5 of 7 — Higher-Order Derivatives

What Are Higher-Order Derivatives?

The second derivative is the derivative of the derivative:

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

Similarly for the third derivative and beyond.

Physical Interpretation

Worked Example 1

Find $f''(x)$ for $f(x) = x^5 - 3x^3 + 2x$

Worked Example 2

Find $\frac{d^2y}{dx^2}$ for $y = e^{2x}$

Pattern: $\frac{d^n}{dx^n}e^{2x} = 2^n e^{2x}$

Find Higher-Order Derivatives 🎯

Concavity and the Second Derivative

The second derivative tells us about concavity:

• $f''(x) > 0$: graph is concave up (holds water, like a cup)
• $f''(x) < 0$: graph is concave down (spills water, like a hill)
• $f^{\prime\prime}(x) = 0$: possible inflection point (concavity may change)

Worked Example 3

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

$f''(x) = 6x$. Concave up when $6x > 0$, i.e., $x > 0$.

AP Tip: The relationship between $f$, $f'$, and $f''$ is tested extensively. Know what each tells you about the graph.

Second Derivative Applications 🎯

Key Takeaways — Part 5

1. Second derivative = derivative of the derivative
2. Physical meaning: position → velocity → acceleration
3. Concavity: $f'' > 0$ = concave up, $f'' < 0$ = concave down
4. Inflection points occur where $f''$ changes sign
5. Second Derivative Test: at critical points, $f'' > 0$ = local min, $f'' < 0$ = local max

📐 Problem-Solving Workshop

Part 6 of 7 — Mixed Differentiation Problems

Choosing the Right Rule

Identify and Apply 🎯

AP-Style Free Response Setup 🎯

Workshop Complete!

You can now:

• Choose the right differentiation rule for any situation
• Combine multiple rules in a single problem
• Apply derivatives to motion problems

📐 Review & Applications

Part 7 of 7 — Comprehensive Review

Complete Derivative Reference

Special Derivatives

Comprehensive Assessment 🎯

Final Problems 🎯

Basic Differentiation Rules — Complete! ✅

You have mastered:

• ✅ Power Rule (including negative/fractional exponents)
• ✅ Product Rule
• ✅ Quotient Rule
• ✅ All six trig derivatives
• ✅ Higher-order derivatives and their applications
• ✅ Combining multiple rules

Ready to move on to the Chain Rule!