Problem Solving Sessions

Master complex derivative problems with step-by-step solutions and expert strategies

📝 Step-by-Step Solutions 🎯 Expert Strategies 💡 Pro Tips ⚡ Quick Methods

📋 Problem Sessions Index

⛓️

Chain Rule Mastery

Complex composite functions

Beginner
✖️

Product & Quotient Rules

Advanced product/quotient problems

Intermediate
🎯

Implicit Differentiation

Solving implicit equations

Intermediate
📐

Higher Order Derivatives

2nd, 3rd, and nth derivatives

Advanced
🎨

Related Rates

Real-world applications

Advanced
🚀

Optimization Problems

Maxima and minima

Expert

Session 1: Chain Rule Mastery

25 min 6 Problems Beginner

🎯 Learning Objectives

  • Identify composite functions in complex expressions
  • Apply chain rule to nested compositions
  • Simplify chain rule results effectively
  • Recognize when to use chain rule vs other rules

Problem 1: Basic Chain Rule

Find the derivative of: f(x) = (3x² + 2x)⁴

Step-by-Step Solution:

1
Identify Inner and Outer Functions
Outer function: f(u) = u⁴
Inner function: u = g(x) = 3x² + 2x
2
Find Derivatives Separately
f'(u) = 4u³
g'(x) = 6x + 2
3
Apply Chain Rule
f'(x) = f'(g(x)) · g'(x)
f'(x) = 4(3x² + 2x)³ · (6x + 2)
4
Simplify
f'(x) = 4(6x + 2)(3x² + 2x)³
f'(x) = 8(3x + 1)(3x² + 2x)³

Final Answer:

f'(x) = 8(3x + 1)(3x² + 2x)³

💡 Pro Tip:

Always factor constants when possible for cleaner results. Here we factored 2 from (6x + 2).

Practice Similar Problem →

Problem 2: Multiple Chain Rule Applications

Find the derivative of: f(x) = sin(cos(x²))

Step-by-Step Solution:

1
Identify Nested Functions
f(x) = sin(u) where u = cos(v) and v = x²
Three layers: sin → cos → x²
2
Apply Chain Rule Multiple Times
f'(x) = cos(cos(x²)) · (-sin(x²)) · (2x)
3
Simplify
f'(x) = -2x · sin(x²) · cos(cos(x²))

Final Answer:

f'(x) = -2x sin(x²) cos(cos(x²))

💡 Pro Tip:

For nested functions, work from the outside in. Each layer adds another multiplication to the chain.

Session 2: Product & Quotient Rules

30 min 8 Problems Intermediate

Problem 1: Product Rule with Chain Rule

Find the derivative of: f(x) = x³ · e^(2x)

Step-by-Step Solution:

1
Identify Product Components
u = x³, v = e^(2x)
u' = 3x²
v' = e^(2x) · 2 = 2e^(2x) (Chain Rule)
2
Apply Product Rule
f'(x) = u'v + uv'
f'(x) = (3x²)(e^(2x)) + (x³)(2e^(2x))
3
Factor Common Terms
f'(x) = e^(2x)(3x² + 2x³)
f'(x) = x²e^(2x)(3 + 2x)

Final Answer:

f'(x) = x²e^(2x)(3 + 2x)
Practice Product Rule →

Session 3: Implicit Differentiation

35 min 7 Problems Intermediate

Problem 1: Circle Equation

Find dy/dx for: x² + y² = 25

Step-by-Step Solution:

1
Differentiate Both Sides with Respect to x
d/dx[x²] + d/dx[y²] = d/dx[25]
2
Apply Chain Rule to y²
2x + 2y · dy/dx = 0
3
Solve for dy/dx
2y · dy/dx = -2x
dy/dx = -x/y

Final Answer:

dy/dx = -x/y

💡 Pro Tip:

Remember that y is a function of x, so d/dx[y] = dy/dx and d/dx[y²] = 2y · dy/dx

Practice Implicit Differentiation →

🎮 Interactive Problem Solver

Try Solving Yourself

Problem:

Find f'(x) for: f(x) = ln(x² + 1)

Your Solution:

🧠 Problem Solving Strategies

🔍

1. Pattern Recognition

Learn to recognize common function patterns: polynomials, trigonometric, exponential, logarithmic, and their combinations.

  • Polynomials: Use power rule
  • Trig functions: Know derivatives of sin, cos, tan
  • Exponentials: e^x is special
  • Logarithms: Remember derivative formulas
⚙️

2. Rule Selection

Choose the right differentiation rule based on function structure:

  • Single function: Basic rules
  • f(g(x)): Chain rule
  • f(x)·g(x): Product rule
  • f(x)/g(x): Quotient rule
  • Multiple rules: Combine as needed
📝

3. Step-by-Step Approach

Never skip steps when learning:

  • Write down each function component
  • Apply rules systematically
  • Simplify after each major step
  • Check your work at each stage
🎯

4. Verification Methods

Always verify your solutions:

  • Use our calculator to check
  • Test with sample x-values
  • Check dimensions/units
  • Look for common error patterns

📥 Download Session Materials

📚 Continue Your Learning

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Visual Explanations

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Practice Problems

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Advanced Calculator

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Learning Center

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