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.

Implicit Differentiation

This section introduces the concept of implicit differentiation, using the example function cos$x² + y$ + 3xy = 1.

Definition: Implicit differentiation is a method used to find the derivative of a function that is not explicitly defined in terms of the independent variable.

The solution demonstrates the process of differentiating both sides of the equation with respect to x, treating y as a function of x. The steps involve applying the chain rule and the product rule, and then solving for dy/dx.

Example: For the function cos$x² + y$ + 3xy = 1, the implicit derivative is: dy/dx = $2x sin(x² + y$ - 3y) / $-sin(x² + y$ + 3x)

This example showcases the power of implicit differentiation in handling complex relationships between variables that are not easily expressed as explicit functions.

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.

Application: Finding Critical Points

This section applies derivative concepts to find the critical points of the function f$x$ = x³ - 6x + 5.

The process involves:

1. Calculating the first derivative: f'$x$ = 3x² - 6
2. Finding critical points by setting f'$x$ = 0 and solving for x
3. Analyzing the behavior of the function around these points using the first derivative test

Definition: Critical points are points where the derivative of a function is zero or undefined, potentially indicating local maxima, minima, or inflection points.

The solution identifies two critical points: x = -√2 ≈ -1.41 and x = √2 ≈ 1.41. Further analysis determines that:

• $-1.41, 10.65$ is a local maximum point
• $1.41, -0.46$ is a local minimum point

Highlight: Understanding how to find and analyze critical points is crucial in optimization problems across various fields, including economics and engineering.

Second Derivative and Inflection Points

This section extends the analysis of the function f$x$ = x³ - 6x + 5 to include the second derivative and identification of inflection points.

The process involves:

1. Calculating the second derivative: f"$x$ = 6x
2. Finding potential inflection points by setting f"$x$ = 0
3. Analyzing the concavity of the function using the second derivative

Definition: An inflection point is a point on a curve at which the curvature changes from concave upwards to concave downwards, or vice versa.

The solution identifies the inflection point at x = 0. The function's behavior is analyzed:

• For x < 0, the function is concave downward
• For x > 0, the function is concave upward
• At x = 0, the inflection point occurs at (0, 5)

Example: The inflection point $0, 5$ marks the transition from concave downward to concave upward behavior in the function f$x$ = x³ - 6x + 5.

A GeoGebra graph is provided to visualize the function, its critical points, and the inflection point. This graphical representation helps in understanding the overall behavior of the function and how it relates to its derivatives.

Highlight: The ability to identify and analyze inflection points is crucial in understanding the shape and behavior of complex functions, with applications in various fields such as physics and economics.

Extrema and Inflection Points - Part 1

This section identifies maximum and minimum points.

Definition: Local extrema occur at points where the function changes from increasing to decreasing or vice versa

Example: Maximum point at $-√2, 10.65$ and minimum point at $√2, -0.46$

Extrema and Inflection Points - Part 2

The section covers second derivative analysis for inflection points.

Definition: Inflection points occur where the concavity of the function changes

Example: Inflection point found at (0,5)

Graphical Representation

The final section provides complete graphical visualization.

Highlight: GeoGebra plot shows all critical points, extrema, and inflection points

Example: Visual confirmation of calculated points: A$-1.41, 10.65$, B$1.41, -0.46$, C(0,5)

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.