Master Derivatives in 30 Days

Core Concept: The derivative represents the instantaneous rate of change - essentially the slope of a curve at a single point.

Daily Tasks:

• Study the geometric meaning: tangent line slope
• Practice calculating derivatives from first principles for simple functions like f(x) = x²
• Understand limits and why they're necessary
• Complete 10 limit problems to build algebraic fluency

The Power Rule is your most-used tool. Master it completely!

Key Rules to Learn:

1. Constant Rule: d/dx[c] = 0
2. Power Rule: Bring down the exponent, reduce exponent by 1
3. Constant Multiple: d/dx[c·f(x)] = c·f'(x)
4. Sum/Difference: Differentiate term by term

Practice Problems (20+ required):

• Polynomials: d/dx[3x⁵ - 2x³ + 7x - 4]
• Fractional exponents: d/dx[√x] = d/dx[x^(1/2)]
• Negative exponents: d/dx[1/x²] = d/dx[x^(-2)]

Memorize these fundamental derivatives - they appear in 90% of calculus problems!

Trigonometric Derivatives:

Exponential & Logarithmic:

Daily Practice:

• Create flashcards for all derivative formulas
• Practice 15-20 mixed problems daily
• Use our derivative calculator to verify your work
• Focus on WHY each rule works, not just memorization

When to use: Multiplying two functions together

Memory Aid: "First times derivative of second, PLUS second times derivative of first"

Example: d/dx[x²·sin(x)]

• First function: f(x) = x², so f'(x) = 2x
• Second function: g(x) = sin(x), so g'(x) = cos(x)
• Answer: 2x·sin(x) + x²·cos(x)

When to use: Dividing one function by another

Memory Aid: "Low d-High minus High d-Low, all over Low-Low (and away we go!)"

Example: d/dx[x³/(x+1)]

• High (numerator): f(x) = x³, so f'(x) = 3x²
• Low (denominator): g(x) = x+1, so g'(x) = 1
• Answer: [(x+1)(3x²) - (x³)(1)] / (x+1)²
• Simplified: (2x³ + 3x²) / (x+1)²

Most Important Rule! Used for composite (nested) functions

Think: Derivative of outer × derivative of inner

Examples:

1. Simple Chain: d/dx[(x² + 1)⁵]

• Outer: ( )⁵, Inner: x² + 1
• Answer: 5(x² + 1)⁴ · 2x = 10x(x² + 1)⁴

2. Trig Chain: d/dx[sin(3x)]

• Outer: sin( ), Inner: 3x
• Answer: cos(3x) · 3 = 3cos(3x)

3. Exponential Chain: d/dx[e^(x²)]

• Outer: e^( ), Inner: x²
• Answer: e^(x²) · 2x = 2xe^(x²)

Advanced Practice: Triple nested functions like d/dx[sin²(e^(3x))]

Use our Chain Rule Calculator to see step-by-step breakdowns of complex chains.

When to use: When y is NOT isolated (e.g., x² + xy + y² = 1)

Key Concept: Treat y as a function of x, so d/dx[y] = dy/dx

Step-by-Step Process:

1. Differentiate both sides with respect to x
2. Every time you differentiate a y-term, multiply by dy/dx
3. Collect all dy/dx terms on one side
4. Factor out dy/dx and solve

Example: Find dy/dx for x² + xy = y³

• d/dx[x²] + d/dx[xy] = d/dx[y³]
• 2x + (y + x·dy/dx) = 3y²·dy/dx
• 2x + y = 3y²·dy/dx - x·dy/dx
• 2x + y = dy/dx(3y² - x)
• Answer: dy/dx = (2x + y)/(3y² - x)

When to use: Variable exponents (e.g., y = x^x)

Technique:

1. Take ln of both sides
2. Use log properties to simplify
3. Differentiate implicitly
4. Solve for dy/dx

Example: y = x^x

• ln(y) = ln(x^x) = x·ln(x)
• (1/y)·dy/dx = ln(x) + x·(1/x) = ln(x) + 1
• dy/dx = y(ln(x) + 1) = x^x(ln(x) + 1)

Concept: The derivative of a derivative

• f'(x) = first derivative (slope/velocity)
• f''(x) = second derivative (concavity/acceleration)
• f'''(x) = third derivative (jerk in physics)

Applications:

• f''(x) > 0 → concave up (smiling curve)
• f''(x) < 0 → concave down (frowning curve)
• f''(x) = 0 → possible inflection point

Definition: How one rate of change relates to another

Universal Strategy:

1. Draw a diagram and label variables
2. Write the geometric equation connecting variables
3. Differentiate with respect to time (t)
4. Substitute known values and solve for unknown rate

Example Types:

• Ladder sliding down wall
• Shadow lengthening
• Water filling/draining tanks
• Balloon expanding

Goal: Find maximum or minimum values

5-Step Method:

1. Identify the quantity to optimize (Area, Volume, Cost, etc.)
2. Write the constraint equation
3. Express everything as a function of ONE variable
4. Find f'(x) = 0 to get critical points
5. Use Second Derivative Test or endpoints to verify max/min

Classic Problems:

• Maximum area of rectangle with fixed perimeter
• Minimum surface area of cylinder with fixed volume
• Closest point on a curve to a given point
• Maximum profit/revenue problems

Complete Function Analysis:

1. Use f(x): Find domain, intercepts, asymptotes
2. Use f'(x): Find increasing/decreasing intervals, local max/min
3. Use f''(x): Find concavity, inflection points

First Derivative Test:

• f'(x) > 0 → function increasing
• f'(x) < 0 → function decreasing
• f'(x) = 0 → critical point (possible max/min)

Second Derivative Test:

• At critical point: f''(x) > 0 → local minimum
• At critical point: f''(x) < 0 → local maximum

Congratulations! Today is about synthesis and confidence building.

Final Day Checklist:

• ✅ Take a full-length timed practice exam (2-3 hours)
• ✅ Review ALL mistakes - understand WHY you made them
• ✅ Redo the 5 hardest problems until perfect
• ✅ Test yourself on all derivative formulas from memory
• ✅ Celebrate your achievement! 🎉