Learning Media: Derivatives Made Easy

🎯 1. Understanding Derivative Concepts

💡 Easy Understanding

A derivative is the instantaneous rate of change of a function. Think of it like the speed of a car at a specific moment!

Limit Definition:

$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$

📝 Example Problem

Problem: Find the derivative of $f(x) = x^2$ using the limit definition!

Solution:

$f'(x) = \lim_{h \to 0} \frac{(x+h)^2 - x^2}{h}$

$= \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - x^2}{h}$

$= \lim_{h \to 0} \frac{2xh + h^2}{h}$

$= \lim_{h \to 0} (2x + h) = 2x$

🧠 Higher-Order Thinking

Problem 1 (Analysis):

If $f(x) = ax^2 + bx + c$ and $f'(2) = 8$ , $f'(3) = 12$ , find values of $a$ and $b$ !

Problem 2 (Evaluation):

Why is the derivative of a constant always zero? Provide geometric and algebraic analysis!

🔢 2. Derivatives of Algebraic Functions

📋 Basic Rules

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

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

$\frac{d}{dx}(f(x) \pm g(x)) = f'(x) \pm g'(x)$ (Sum/Difference)

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

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

📝 Example Problems

Example 1:

Find the derivative of $f(x) = 3x^4 - 2x^3 + 5x - 7$

$f'(x) = 12x^3 - 6x^2 + 5$

Example 2:

Find the derivative of $f(x) = (2x + 1)(x^2 - 3)$

Using the product rule:

$f'(x) = 2(x^2 - 3) + (2x + 1)(2x) = 6x^2 + 2x - 6$

🧠 Higher-Order Thinking

Problem 1 (Synthesis):

If $f(x) = \frac{x^3 - 2x^2 + 1}{x^2 + 1}$ , find $f'(1)$ !

Problem 2 (Analysis):

Prove that if $f(x) = x^n$ , then $f^{(n)}(x) = n!$ (nth derivative)

🎯 Tips & Tricks

• For $x^n$ : bring down the power, multiply by original power
• Constants disappear when differentiated
• Product rule: $(uv)' = u'v + uv'$
• Quotient rule: $\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}$
• Chain rule: $(f(g(x)))' = f'(g(x)) \cdot g'(x)$

📐 3. Derivatives of Trigonometric Functions

📋 Basic Formulas

$\frac{d}{dx}(\sin x) = \cos x$

$\frac{d}{dx}(\cos x) = -\sin x$

$\frac{d}{dx}(\tan x) = \sec^2 x$

$\frac{d}{dx}(\cot x) = -\csc^2 x$

$\frac{d}{dx}(\sec x) = \sec x \tan x$

$\frac{d}{dx}(\csc x) = -\csc x \cot x$

💡 Memory Tips

"Co-function" Pattern:

Functions starting with "co" (cos, cot, csc) get negative signs

Square Pattern:

tan → sec², cot → csc²

📝 Example Problems

Example 1:

$f(x) = 3\sin x + 2\cos x$

$f'(x) = 3\cos x - 2\sin x$

Example 2:

$f(x) = \sin(2x + 1)$ (using chain rule)

$f'(x) = \cos(2x + 1) \cdot 2 = 2\cos(2x + 1)$

🧠 Higher-Order Thinking

Problem 1 (Analysis):

If $f(x) = \frac{\sin x}{1 + \cos x}$ , prove that $f'(x) = \frac{1}{1 + \cos x}$ !

Problem 2 (Synthesis):

Find the maximum value of $f'(x)$ if $f(x) = x + 2\sin x$ on interval $[0, 2\pi]$ !

Problem 3 (Evaluation):

Why is $\frac{d}{dx}(\sin x) = \cos x$ ? Provide proof using limit definition!

🚀 4. Applications of Derivatives

🎯 Main Applications

1. Rate of Change

Velocity, acceleration, growth rates

2. Extreme Values

Maximum and minimum of functions

3. Optimization

Real-world max-min problems

4. Tangent Lines

Equations of tangent lines to curves

📝 Application Examples

Example 1 - Rate of Change:

If position of object $s(t) = 2t^3 - 6t^2 + 4t$ , find:

a) Velocity at $t = 2$

b) Acceleration at $t = 1$

$v(t) = s'(t) = 6t^2 - 12t + 4$

$a(t) = v'(t) = 12t - 12$

a) $v(2) = 24 - 24 + 4 = 4$

b) $a(1) = 12 - 12 = 0$

Example 2 - Tangent Line:

Find the equation of tangent line to curve $y = x^2 - 4x + 3$ at point $(2, -1)$

$y' = 2x - 4$

Slope at $x = 2$ : $m = 2(2) - 4 = 0$

Equation: $y - (-1) = 0(x - 2)$

Therefore: $y = -1$

🧠 Higher-Order Thinking

Problem 1 (Optimization):

A farmer has 100 m of wire to make a rectangular pen. Find the dimensions for maximum area!

Problem 2 (Motion Analysis):

A ball is thrown upward with height $h(t) = -5t^2 + 20t + 2$ . When does it reach maximum height and what is that height?

Problem 3 (Related Rates):

An inverted conical tank with radius 3 m and height 6 m is being filled at 2 m³/min. How fast is the water level rising when the water is 4 m deep?

📊 5. Graph Analysis Using Derivatives

📋 Systematic Steps

1. Domain & Range

Determine the domain and range

2. Intercepts

Find x and y intercepts

3. First Derivative

Find increasing/decreasing intervals and critical points

4. Second Derivative

Find concavity and inflection points

5. Asymptotes

Vertical, horizontal, and oblique

💡 Analysis Tips

$f'(x) > 0$ : Function is increasing

$f'(x) < 0$ : Function is decreasing

$f''(x) > 0$ : Concave up

$f''(x) < 0$ : Concave down

📝 Example Analysis

Analyze $f(x) = x^3 - 3x^2 + 2$ :

$f'(x) = 3x^2 - 6x = 3x(x-2)$

Critical points: $x = 0, x = 2$

$f''(x) = 6x - 6 = 6(x-1)$

Inflection point: $x = 1$

• $x < 0$ : increasing, concave down

• $0 < x < 1$ : decreasing, concave down

• $1 < x < 2$ : decreasing, concave up

• $x > 2$ : increasing, concave up

🧠 Higher-Order Thinking

Problem 1 (Graph Analysis):

Sketch the graph of $f(x) = \frac{x^2-4}{x-1}$ with complete analysis!

Problem 2 (Synthesis):

Find values of $a$ and $b$ so that $f(x) = ax^3 + bx^2 + 12x - 5$ has an inflection point at $(1, 4)$ !

Problem 3 (Evaluation):

Explain why inflection points occur when $f''(x) = 0$ and how to distinguish them from extreme points!

🎨 Graph Visualization

🎯 Interactive Quiz - Test Your Understanding

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Quiz Instructions:

• 15 multiple choice questions covering all derivative topics
• Each correct answer earns 1 point
• Explanations appear after answering
• Click "Next Question" to continue

Basic Concepts Question 1 of 15

🎯 1. Understanding Derivative Concepts

💡 Easy Understanding

Limit Definition:

📝 Example Problem

🧠 Higher-Order Thinking

🔢 2. Derivatives of Algebraic Functions

📋 Basic Rules

📝 Example Problems

🧠 Higher-Order Thinking

🎯 Tips & Tricks

📐 3. Derivatives of Trigonometric Functions

📋 Basic Formulas

💡 Memory Tips

📝 Example Problems

🧠 Higher-Order Thinking

🚀 4. Applications of Derivatives

🎯 Main Applications

1. Rate of Change

2. Extreme Values

3. Optimization

4. Tangent Lines

📝 Application Examples

🧠 Higher-Order Thinking

📊 5. Graph Analysis Using Derivatives

📋 Systematic Steps

1. Domain & Range

2. Intercepts

3. First Derivative

4. Second Derivative

5. Asymptotes

💡 Analysis Tips

📝 Example Analysis

🧠 Higher-Order Thinking

🎨 Graph Visualization

🎯 Interactive Quiz - Test Your Understanding

Score: 0

Correct: 0

Question: 1/15

Quiz Instructions:

What is the limit definition of the derivative of function f(x)?

💡 Explanation:

🎉 Quiz Complete!