Rules of Differentiation

Basic Differentiation Rules

Differentiation is a fundamental operation in calculus, used to find the rate at which a function changes. The following are the primary rules for differentiating functions:

• Constant Rule: If (where is a constant), then Example:

• Power Rule: If , then Example:

• Constant Multiple Rule: If , then Example:

• Sum and Difference Rule: If , then Example:

Common Derivatives Table

Product Rule

Formula and Application

The product rule is used when differentiating the product of two functions:

• Formula: If , then

• Memory Device: "First times derivative of second, PLUS second times derivative of first"

• Step-by-Step Process:

  1. Identify the two functions being multiplied: and

  2. Find and

  3. Apply the formula

• Example: , ,

Quotient Rule

Formula and Application

The quotient rule is used when differentiating the quotient of two functions:

• Formula: If , then

• Memory Device: "Lo D Hi minus Hi D Lo, over Lo squared" - Lo = denominator (bottom function) - Hi = numerator (top function) - D = derivative

• Step-by-Step Process:

  1. Identify numerator and denominator

  2. Find and

  3. Apply the formula

• Example: , ,

Chain Rule

Formula and Application

The chain rule is used to differentiate composite functions, where one function is nested inside another.

• Formula: If , then

• Alternative Notation: where and

• Memory Device: "Derivative of outside times derivative of inside"

• Step-by-Step Process:

  1. Identify the outer function and inner function

  2. Find the derivative of the outer function, leaving the inside unchanged

  3. Multiply by the derivative of the inner function

• Example: Outer function: , inner function:

Implicit Differentiation

Key Principle and Steps

Implicit differentiation is used when a function is not given explicitly as , but rather as a relation involving and .

• Key Principle: When differentiating terms with , use chain rule and multiply by

• Step-by-Step Process:

  1. Differentiate both sides of the equation with respect to

  2. For terms with only , differentiate normally

  3. For terms with , differentiate and multiply by

  4. For terms with both and , use product rule or chain rule

  5. Collect all terms with on one side

  6. Factor out and solve

• Example: Differentiate both sides:

Related Rates

General Strategy and Applications

Related rates problems involve rates of change of related quantities, often with respect to time. These are common in geometric and motion scenarios.

• General Strategy:

  1. Read carefully and identify all variables

  2. Identify what rates are given and what you need to find

  3. Draw a diagram if helpful (especially for geometric problems)

  4. Write an equation relating the variables

  5. Differentiate both sides with respect to time ()

  6. Substitute known values and solve for the unknown rate

• Common Scenarios:

  • Areas and volumes changing

  • Similar triangles

  • Pythagorean theorem applications

  • Motion problems (objects moving toward/away from each other)

  • Shadow problems

  • Ladder sliding problems

• Example: Expanding Circle Given: ft/min, ft Find: when Equation: Differentiate: Substitute: ft2/min

Practice Problems & Solutions

Sample Problems

• Product Rule Practice: Solution:

• Quotient Rule Practice: Solution:

• Chain Rule Practice: Solution:

• Implicit Differentiation Practice: Solution:

• Related Rates Practice: A balloon is spherical and expanding. If the volume increases at $100r = 5V = \frac{4}{3}\pi r^3dV/dt = 4\pi r^2 dr/dtdr/dt = \frac{dV/dt}{4\pi r^2} = \frac{100}{4\pi (5^2)} = \frac{100}{100\pi} = \frac{1}{\pi}$ cm/s

Exam Strategies & Tips

Preparation and Test-Taking

• Before the Exam:

  • Review formulas the night before, then get good sleep

  • Pack extra calculator batteries and arrive early

  • Identify your strengths – tackle familiar problems first

• During the Exam:

  • Read each problem twice before starting

  • Show all work – partial credit is available

  • Write clearly and organize your work

  • Check units in your final answers

• Time Management:

  • Budget your time; don’t spend too long on one problem

  • Two-pass system: do easy problems first, then return to harder ones

  • Leave time to review answers

• Calculator Usage:

  • Know when to use a calculator, when to leave expressions exact

  • Use the step-by-step calculator guide

  • Double-check entries for complex expressions

Common Mistakes to Avoid

Frequent Errors in Differentiation

• Product Rule Mistakes:

  • X DON’T:

  • DO:

• Quotient Rule Mistakes:

  • X DON’T: Mix up order in the numerator

  • DO: Remember "Lo D Hi minus Hi D Lo over Lo squared"

• Implicit Differentiation Mistakes:

  • X DON’T: Forget to multiply by when differentiating terms

  • DO: Remember every term needs factor

• Related Rates Mistakes:

  • X DON’T: Substitute values before differentiating

  • DO: Set up equations, differentiate, then substitute

• General Mistakes:

  • X DON’T: Simplify too early – keep expressions exact

  • X DON’T: Skip steps – show complete work

  • X DON’T: Forget to check reasonableness of answers

Quick Reference Formulas

Differentiation Rules

• Constant:

• Power:

• Product:

• Quotient:

• Chain:

Common Derivatives

Related Rates Strategy

• Identify variables and rates

• Draw diagram if needed

• Write relating equation

• Differentiate with respect to time

• Substitute and solve

Good luck on your midterm! Remember to: Set up each problem carefully, show all work, and check your answers for reasonableness.