Calc Notes 2/21

Understanding Derivatives

• Derivatives are used to analyze how functions change and identify points of stability, like maximum or minimum values.

• In real mathematics and physics, functions are often presented simply (e.g., x²), which may limit how we discuss their behaviors.

Diagrammatic Approach

• Visual representation of functions allows for better understanding of derivatives.

• Derivatives are treated as operations that functions can perform, unlike numbers which are static points.

Chain Rule

• The chain rule is a derivative rule useful for composite functions, highlighting the relationship between inner and outer functions.

• Example: For a composite function such as f(g(x)), the derivative is found by applying the chain rule.

Decompositions

• Decomposition involves breaking down complex functions into polynomial forms for easier differentiation.

• Example: If you have a polynomial of degree 14, identify the inner and outer functions to simplify the process.

Identifying Functions

• It is essential to determine which part of a function acts first when plugging in variables.

• The 'inside' function is typically the first operation applied to the variable (e.g., for e^x, the 'inside' may be the exponent).

• Use terminology like 'inside' to understand the order of operations in functions as an abstraction.

Relationship of Functions

• Functions have a hierarchical relationship allowing for layers of complexity, where the function's output can be the input to another.

• Clarifying what operations apply to what variables gives a clearer understanding of the overall function.

Practical Examples

• When differentiating composite functions like ( f(g(x)) ), evaluate the outer derivative first, followed by the inner derivative where appropriate.

• Example: If f'(x) is cosine and g is a polynomial, derive carefully and compose step by step.

Importance of Order of Operations

• Being clear about the order of operations prevents mistakes when composing and differentiating functions.

• Consider a situation where you have sin(x²); recognize the square takes precedence before the sine function applies.

Variable Representation

• Variables in derivatives are simply placeholders representing whatever input you use—they are not fixed values.

• This is crucial to keep in mind while working with composite functions.

Exploring Logarithmic Functions

• Logarithms can help simplify differentiation when dealing with exponential functions.

• Use log properties to reframe exponential derivatives, significantly simplifying calculations.

Derivative Formulas

• There's a distinction in notation using primes for derivatives. d/dx format is also commonly used.

• Acknowledge different derivative notations as tools for clarity, but remember they convey similar concepts.

Multi-layered Derivatives

• Examine functions with several nested layers (e.g., sin(cos(tan(x)))—approach it by working from the outer layer to the inner layers, applying each derivative step by step.

• Recognize the complexity adds significantly more steps, requiring careful attention to the order of derivatives.