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.