BackDifferentiation Rules and Applications in Calculus
Study Guide - Smart Notes
Tailored notes based on your materials, expanded with key definitions, examples, and context.
Chain Rule and Composite Functions
Definition and Application
The chain rule is a fundamental technique in calculus for differentiating composite functions. If a function y is defined as the composition of two functions, y = f(g(x)), the derivative is found by multiplying the derivative of the outer function evaluated at the inner function by the derivative of the inner function.
Formula:
Example: If , then
Basic Differentiation Rules
Power, Product, and Quotient Rules
These rules are essential for differentiating algebraic expressions.
Power Rule:
Product Rule:
Quotient Rule:
Example:
Derivatives of Trigonometric Functions
Standard Trigonometric Derivatives
The derivatives of trigonometric functions are frequently used in calculus.
Function | Derivative |
|---|---|
Example:
Derivatives of Inverse Trigonometric Functions
Formulas and Applications
Inverse trigonometric functions have specific differentiation rules.
Function | Derivative |
|---|---|
Derivatives of Exponential and Logarithmic Functions
Exponential Functions
General Exponential:
Natural Exponential:
Logarithmic Functions
General Logarithm:
Natural Logarithm:
Example:
Special Algebraic Forms
Square of Binomial
Formula:
Worked Examples
Applying Differentiation Rules
Example 1:
Example 2:
Example 3:
Example 4:
Summary Table: Common Derivatives
Function | Derivative |
|---|---|
Additional info:
Some formulas and examples were inferred and expanded for clarity and completeness.
Tables were reconstructed from the images and text for better readability.
