Derivative Calculation Using Limit Definition

This section demonstrates the process of calculating the derivative of a function using the limit definition. The example function f(x) = 5 + 4x - 3x² is used to illustrate the step-by-step procedure.

Definition: The derivative of a function is defined as the limit of the difference quotient as h approaches zero: f'(x) = lim[h→0] (f(x+h) - f(x)) / h

The solution involves expanding f(x+h), subtracting f(x), simplifying, and then taking the limit as h approaches zero. The final result shows that the derivative of the function f(x) = 5 + 4x - 3x² is f'(x) = -6x + 4.

Example: For f(x) = 5 + 4x - 3x², the derivative calculation process yields f'(x) = -6x + 4.

A graphical representation using GeoGebra is provided to visualize the function and its derivative. The graph shows the original function, its tangent line at a specific point, and the slope of the tangent line, which corresponds to the derivative value at that point.

Highlight: The use of GeoGebra allows for a visual understanding of the relationship between a function and its derivative, enhancing the learning experience for students studying calculus and derivatives.

Higher-Order Derivatives

This section delves into the calculation of higher-order derivatives, using the function f(x) = ln(x² + x³ - 2) as an example.

The process involves calculating the first, second, and third derivatives of the given function. Each step is explained in detail, showcasing the application of various derivative rules and techniques.

Highlight: Higher-order derivatives are essential in understanding the behavior of functions, particularly in areas such as optimization and Taylor series expansions.

The solutions for each derivative are provided:

1. First derivative: f'(x) = (2x + 3x²) / (x² + x³ - 2)
2. Second derivative: f"(x) = (5x³ - 20x² - 10) / (x² + x³ - 2)²
3. Third derivative: A complex expression is derived (too long to include here)

A GeoGebra graph is included to visualize the original function and its first three derivatives on the same plane, allowing for a comparative analysis of their behaviors.

Example: The GeoGebra graph shows how the original function and its derivatives relate to each other, providing valuable insights into the function's rate of change at different orders.

Derivative Rules and Logarithmic Functions

This section focuses on applying derivative rules to more complex functions, particularly those involving logarithms. The example function provided is f(x) = ln(2 + √(5x+6)).

Vocabulary: ln refers to the natural logarithm, which is the logarithm with base e (Euler's number).

The solution demonstrates the step-by-step application of derivative rules, including the chain rule and the derivative of the natural logarithm. The final derivative is expressed as:

f'(x) = 5 / (2(2 + √(5x+6)) √(5x+6))

A GeoGebra graph is included to visualize the function and its derivative, showing the tangent line at a specific point and its slope.

Highlight: Understanding the derivatives of logarithmic functions is crucial in many areas of calculus and its applications in science and engineering.