Derivative Troubleshooting Guide

Diagnose and fix common derivative calculation errors with step-by-step solutions

🔧 Fix Common Errors 🎯 Step-by-Step Solutions 📊 Visual Diagnostics ⚡ Quick Fixes

🔍 Error Diagnostic Tool

What's Your Error?

⛓️ Chain Rule Troubleshooting

Error: Forgetting to Multiply by Inner Derivative

Common

❌ Incorrect Solution

Function: f(x) = sin(x²)

Student's answer: f'(x) = cos(x²)

✅ Correct Solution

f(x) = sin(x²)

f'(x) = cos(x²) · 2x

f'(x) = 2x cos(x²)

🔬 Diagnosis

The student correctly identified the outer function (sin) but forgot to multiply by the derivative of the inner function (x²).

💡 Solution Steps

  1. Identify inner function: g(x) = x²
  2. Identify outer function: f(u) = sin(u)
  3. Apply chain rule: f'(x) = f'(g(x)) · g'(x)
  4. Calculate: f'(x) = cos(x²) · 2x
  5. Simplify: f'(x) = 2x cos(x²)

🛡️ Prevention Tips

  • Always write out both functions separately
  • Use parentheses to clearly show composition
  • Check: "Did I multiply by the derivative of the inside?"
Practice Chain Rule Calculator →

Error: Misidentifying Inner and Outer Functions

Intermediate

❌ Incorrect Solution

Function: f(x) = (sin x)²

Student's answer: f'(x) = 2x sin(x)

✅ Correct Solution

f(x) = (sin x)² = sin²(x)

Let u = sin x, then f(u) = u²

f'(x) = 2u · u' = 2 sin x · cos x

f'(x) = 2 sin x cos x = sin(2x)

💡 Solution Steps

  1. Rewrite: f(x) = [sin(x)]²
  2. Inner function: g(x) = sin(x)
  3. Outer function: f(u) = u²
  4. g'(x) = cos(x)
  5. f'(x) = 2·sin(x)·cos(x)
  6. Use identity: 2 sin x cos x = sin(2x)

✖️ Product Rule Troubleshooting

Error: Adding Instead of Following Product Rule

Common

❌ Incorrect Solution

Function: f(x) = x · sin(x)

Student's answer: f'(x) = 1 + cos(x)

✅ Correct Solution

Let u = x, v = sin(x)

u' = 1, v' = cos(x)

f'(x) = u'v + uv'

f'(x) = 1·sin(x) + x·cos(x)

f'(x) = sin(x) + x cos(x)

🔬 Diagnosis

The student added the derivatives instead of using the product rule formula.

🎵 Helpful Mnemonic

"Derivative of First times Second,
Plus First times Derivative of Second"

Practice Product Rule Calculator →

➗ Quotient Rule Troubleshooting

Error: Getting Signs Wrong in Numerator

Very Common

❌ Incorrect Solution

Function: f(x) = (x² + 1)/(x - 1)

Student's answer: f'(x) = [2x·(x-1) + (x²+1)·1] / (x-1)²

✅ Correct Solution

Let u = x² + 1, v = x - 1

u' = 2x, v' = 1

f'(x) = (u'v - uv') / v²

f'(x) = [2x·(x-1) - (x²+1)·1] / (x-1)²

f'(x) = (2x² - 2x - x² - 1) / (x-1)²

f'(x) = (x² - 2x - 1) / (x-1)²

🎵 Helpful Mnemonic

"Low D-High minus High D-Low,
Over the Square of What's Below"

📝 Sign & Parentheses Troubleshooting

Error: Distributing Negative Signs Incorrectly

Common

❌ Incorrect Solution

Function: f(x) = 3x² - 4x + 5

Student's answer: f'(x) = 6x - 4 + 0 = 6x - 4

But then: f''(x) = 6 - 0 = 6

Error: Forgot negative sign in second derivative

✅ Correct Solution

f(x) = 3x² - 4x + 5

f'(x) = 6x - 4

f''(x) = 6

Note: -4 differentiates to 0, not -0

💡 Important Tip

When differentiating polynomials, handle each term separately. Constants become 0, and coefficients carry through with proper signs.

Error: Missing Parentheses in Chain Rule

Common

❌ Incorrect Solution

Function: f(x) = e^(2x + 3)

Student's answer: f'(x) = e^2x + 3 · 2

✅ Correct Solution

f(x) = e^(2x + 3)

Let u = 2x + 3

f'(x) = e^(2x + 3) · 2

f'(x) = 2e^(2x + 3)

💡 Parentheses Rule

Always use parentheses when the exponent contains more than one term. e^(2x+3) is different from e^2x + 3.

🎯 Interactive Error Finder

Find Errors in These Solutions

Exercise 1: Find the error

Function: f(x) = (x² + 1)³

Given solution: f'(x) = 3(x² + 1)² · 2x = 6x(x² + 1)²

Is this correct?

Exercise 2: Find the error

Function: f(x) = x · e^x

Given solution: f'(x) = 1 · e^x = e^x

What's wrong?

✅ Error Prevention Checklist

✓

Before Starting

  • Identify function type (polynomial, trigonometric, etc.)
  • Check for composition (chain rule needed?)
  • Check for products/quotients
✓

During Calculation

  • Write out each step clearly
  • Use parentheses generously
  • Check signs at each step
✓

After Calculation

  • Verify with calculator tool
  • Check domain restrictions
  • Simplify final answer

📚 Additional Help Resources

Visual Explanations

See common errors visualized with interactive graphs and animations.

Problem Solving Sessions

Watch step-by-step solutions to challenging derivative problems.

Calculus Glossary

Review key terms and definitions to build foundational knowledge.

Derivative Rules Reference

Complete reference guide for all differentiation rules with examples.

Practice Problems

Test your troubleshooting skills with categorized practice problems.

Derivative Calculator

Use our advanced calculator to check your work and find errors.