♾️AP Calculus AB/BC Unit 2 – Fundamentals of Differentiation

What's Differentiation Anyway?

• Differentiation calculates the rate of change of a function at a given point
• Determines the slope of the tangent line to a curve at a specific point
• Enables us to analyze how a function changes as its input changes
• Fundamental concept in calculus and has numerous real-world applications
  • Velocity and acceleration in physics
  • Marginal cost and revenue in economics
• Represented by the derivative of a function, denoted as $f'(x)$ for a function $f(x)$
• The derivative is the limit of the difference quotient as the change in $x$ approaches zero:
  • $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$
• Differentiation and integration are inverse operations, forming the foundation of calculus

The Derivative: Your New Best Friend

• The derivative of a function $f(x)$ is another function that gives the slope of the tangent line to the graph of $f(x)$ at any point
• Derivatives allow us to find rates of change, optimize functions, and analyze the behavior of curves
• For a linear function $f(x) = mx + b$, the derivative is the constant slope $m$
• The derivative of a constant function is always zero, as the slope is horizontal
• Power Rule: For a function $f(x) = x^n$, the derivative is $f'(x) = nx^{n-1}$
  • Example: If $f(x) = x^3$, then $f'(x) = 3x^2$
• Derivatives of common functions:
  • $\frac{d}{dx} \sin(x) = \cos(x)$
  • $\frac{d}{dx} \cos(x) = -\sin(x)$
  • $\frac{d}{dx} e^x = e^x$
  • $\frac{d}{dx} \ln(x) = \frac{1}{x}$

Rules of the Game: Differentiation Techniques

• Sum Rule: The derivative of a sum is the sum of the derivatives
  • $\frac{d}{dx} [f(x) + g(x)] = f'(x) + g'(x)$
• Difference Rule: The derivative of a difference is the difference of the derivatives
  • $\frac{d}{dx} [f(x) - g(x)] = f'(x) - g'(x)$
• Constant Multiple Rule: Constants can be factored out when differentiating
  • $\frac{d}{dx} [c \cdot f(x)] = c \cdot f'(x)$, where $c$ is a constant
• Product Rule: For two functions $f(x)$ and $g(x)$, the derivative of their product is:
  • $\frac{d}{dx} [f(x) \cdot g(x)] = f(x) \cdot g'(x) + f'(x) \cdot g(x)$
• Quotient Rule: For two functions $f(x)$ and $g(x)$, the derivative of their quotient is:
  • $\frac{d}{dx} [\frac{f(x)}{g(x)}] = \frac{g(x) \cdot f'(x) - f(x) \cdot g'(x)}{[g(x)]^2}$
• These rules allow us to break down complex functions into simpler components and differentiate them step by step

Tricky Stuff: Chain Rule and Implicit Differentiation

• Chain Rule: Used for differentiating composite functions
  • If $h(x) = f(g(x))$, then $h'(x) = f'(g(x)) \cdot g'(x)$
  • Differentiate the outer function, then multiply by the derivative of the inner function
  • Example: If $h(x) = \sin(x^2)$, then $h'(x) = \cos(x^2) \cdot 2x$
• Implicit Differentiation: Used when a function is not explicitly defined as $y = f(x)$
  • Differentiate both sides of the equation with respect to $x$, treating $y$ as a function of $x$
  • Example: For the equation $x^2 + y^2 = 25$, implicitly differentiating yields:
    • $2x + 2y \cdot \frac{dy}{dx} = 0$
    • Solve for $\frac{dy}{dx}$ to find the derivative
• These techniques are essential for dealing with more complex functions and relationships

Putting It to Work: Applications of Derivatives

• Optimization: Derivatives can help find the maximum or minimum values of a function
  • Set the derivative equal to zero and solve for the critical points
  • Evaluate the function at the critical points and endpoints to find the extrema
• Related Rates: Derivatives allow us to find the rate of change of one quantity with respect to another
  • Example: If the radius of a circle is increasing at a rate of 2 cm/s, how fast is the area changing when the radius is 5 cm?
• Marginal Analysis: Derivatives help analyze the impact of small changes in variables
  • Marginal cost is the derivative of the total cost function
  • Marginal revenue is the derivative of the total revenue function
• Velocity and Acceleration: Derivatives describe the motion of objects
  • Velocity is the derivative of position with respect to time
  • Acceleration is the derivative of velocity with respect to time
• These applications demonstrate the power and versatility of derivatives in solving real-world problems

Graphing with Derivatives: A Visual Journey

• First Derivative Test: Determines the increasing or decreasing behavior of a function
  • If $f'(x) > 0$ on an interval, $f(x)$ is increasing on that interval
  • If $f'(x) < 0$ on an interval, $f(x)$ is decreasing on that interval
• Second Derivative Test: Determines the concavity of a function
  • If $f''(x) > 0$ at a point, the graph is concave up at that point
  • If $f''(x) < 0$ at a point, the graph is concave down at that point
• Inflection Points: Points where the concavity of a function changes
  • Occur where $f''(x) = 0$ or is undefined
• Sketching Curves: Derivatives provide information about the shape and behavior of a function's graph
  • Use the first and second derivative tests to determine increasing/decreasing intervals and concavity
  • Identify local maxima, local minima, and inflection points
  • Plot key points and connect them with curves based on the derivative information
• Visualizing derivatives helps develop a deeper understanding of a function's behavior and characteristics

Common Pitfalls and How to Dodge Them

• Forgetting to use the Chain Rule when differentiating composite functions
  • Always identify the inner and outer functions and apply the Chain Rule
• Misapplying the Product or Quotient Rule
  • Remember to differentiate each function separately and follow the correct formulas
• Incorrectly handling negative exponents when using the Power Rule
  • Subtract 1 from the exponent and multiply by the original exponent, even if it's negative
• Differentiating constants as if they were variables
  • The derivative of a constant is always zero
• Confusing the signs when using the Second Derivative Test
  • $f''(x) > 0$ indicates concave up, while $f''(x) < 0$ indicates concave down
• Overlooking the domain of a function when differentiating
  • Be aware of any restrictions on the domain, such as avoiding division by zero
• Practice, attention to detail, and a solid understanding of the rules and techniques will help avoid these common mistakes

Beyond the Basics: A Peek at Advanced Topics

• L'Hôpital's Rule: Used to evaluate limits of indeterminate forms (0/0, ∞/∞, etc.)
  • If $\lim_{x \to a} \frac{f(x)}{g(x)}$ is an indeterminate form, then $\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}$, provided the limit on the right exists
• Partial Derivatives: Derivatives of functions with multiple variables
  • Differentiate with respect to one variable while treating the others as constants
  • Useful in multivariable calculus and applications such as gradient descent in machine learning
• Parametric Differentiation: Finding derivatives of curves defined by parametric equations
  • If $x = f(t)$ and $y = g(t)$, then $\frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{g'(t)}{f'(t)}$
• Implicit Differentiation in Higher Dimensions: Extending implicit differentiation to functions with multiple variables
  • Useful for finding tangent planes to surfaces in three-dimensional space
• These advanced topics build upon the foundation of basic differentiation and open up new areas of study and application in mathematics and related fields