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Basic Differentiation Rules

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Basic Differentiation Rules - Complete Interactive Lesson

Part 1: The Power Rule

📐 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

Function	Derivative
$x^5$	$5x^4$
$x^{100}$

Constant Multiple Rule

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

Constants just "come along for the ride."

Function	Derivative
$7x^3$	$21x^2$
$-4x^5$

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:

Original	Rewrite	Derivative
$\frac{1}{x^3}$	$x^{-3}$

Negative & Fractional Exponents 🎯

Key Takeaways — Part 1

Power Rule: $\frac{d}{dx}x^n = nx^{n-1}$ for any real

Part 2: Product Rule

📐 The Product Rule

Part 2 of 7 — Product Rule

Why Can't We Just Multiply the Derivatives?

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

Part 3: Quotient Rule

📐 The Quotient Rule

Part 3 of 7 — Quotient Rule

The Quotient Rule

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

Part 4: Trig Derivatives

📐 Trigonometric Derivatives

Part 4 of 7 — Trig Derivatives

The Six Trig Derivatives

Function	Derivative
$\sin x$	$\cos x$
$\cos x$	$-\sin x$

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:

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

Part 6: Mixed Differentiation Problems

📐 Problem-Solving Workshop

Part 6 of 7 — Mixed Differentiation Problems

Choosing the Right Rule

Situation	Rule to Use
Single term: $x^n$	Power Rule
Product: $f \cdot g$	Product Rule
Quotient:

Part 7: Comprehensive Review

📐 Review & Applications

Part 7 of 7 — Comprehensive Review

Complete Derivative Reference

Rule	Formula
Power	$\frac{d}{dx}x^n = nx^{n-1}$

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

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

Constants vanish:

\frac{d}{dx}[c] = 0

Constant multiples pass through:

\frac{d}{dx}[cf(x)] = cf'(x)

Sum/Difference: differentiate term by term

Always rewrite roots and fractions as power expressions first

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

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

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

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

Part	Value
$f = x^2$ , $f' = 2x$	$g = \sin x$ , $g' = \cos x$
Product Rule	$2x \sin x + x^2 \cos x$

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

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

[g(x)]2f′(x)g(x)−f(x)g′(x)

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

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

Part	Value
$f = x^2$ , $f' = 2x$	$g = \sin x$ , $g' = \cos x$
Quotient Rule	$\frac{2x\sin x - x^2\cos x}{\sin^2 x}$

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

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

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

Problem	Solution
$\frac{d}{dx}(3\sin x + 2\cos x)$	$3\cos x - 2\sin x$
$\frac{d}{dx}(x^2 + \tan x)$	$2 x +$
$\frac{d}{dx}(5\sec x)$	$5\sec x \tan x$

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

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

Similarly for the third derivative and beyond.

Physical Interpretation

Derivative	Meaning
$f(t)$	Position
$f'(t)$	Velocity (rate of change of position)
$f''(t)$	Acceleration (rate of change of velocity)
$f^{(3)}(t)$	Jerk (rate of change of acceleration)

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

Step	Result
$f'(x)$	$5x^4 - 9x^2 + 2$
$f''(x)$	$20x^3 - 18x$

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

Step	Result
$y'$	$2e^{2x}$
$y''$	$4e^{2x}$

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

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

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

Function	Derivative
$e^x$	$e^x$
$\ln x$	$\frac{1}{x}$
$a^x$	$a^x \ln a$
$\log_a x$	$\frac{1}{x \ln a}$

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!

cos2xcosx⋅cosx−sinx(−sinx)=

Basic Differentiation Rules

Worked Example

The Product Rule

Worked Example 1

Worked Example 2

When to Use Product Rule vs. Expand

Worked Example 3

Key Takeaways — Part 2

Worked Example 1

Worked Example 2

Deriving Trig Derivatives via Quotient Rule

When to Avoid the Quotient Rule

Key Takeaways — Part 3

Pattern Recognition

Worked Examples

Special Values to Know

Combining Rules

Key Takeaways — Part 4

Physical Interpretation

Worked Example 1

Worked Example 2

Concavity and the Second Derivative

Worked Example 3

Key Takeaways — Part 5

Workshop Complete!

Special Derivatives

Basic Differentiation Rules — Complete! ✅