Differentiation is the cornerstone of calculus, allowing us to analyze how functions change. It's all about finding rates of change and slopes of tangent lines at specific points. This powerful tool has applications in physics, economics, and many other fields.The derivative, denoted as f'(x), is the key player in differentiation. It's a new function that gives the slope of the original function at any point. Understanding derivatives and mastering differentiation techniques opens doors to solving complex real-world problems.

## Study Guides for Unit 2

2.0Unit 2 Overview: Differentiation6 min read

2.1Defining Average and Instantaneous Rates of Change at a Point3 min read

2.2Defining the Derivative of a Function and Using Derivative Notation4 min read

2.3Estimating Derivatives of a Function at a Point4 min read

2.4Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist6 min read

2.5Applying the Power Rule2 min read

2.6Derivative Rules: Constant, Sum, Difference, and Constant Multiple4 min read

2.7Derivatives of cos x, sinx, e^x, and ln x2 min read

2.8The Product Rule3 min read

2.9The Quotient Rule3 min read

2.10Finding the Derivatives of Tangent, Cotangent, Secant, and/or Cosecant Functions3 min read

## 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 $f'(x)$f′(x) for a function $f(x)$f(x)
- The derivative is the limit of the difference quotient as the change in $x$x approaches zero:
- $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$f′(x)=limh→0hf(x+h)−f(x)

- Differentiation and integration are inverse operations, forming the foundation of calculus

## The Derivative: Your New Best Friend

- The derivative of a function $f(x)$f(x) is another function that gives the slope of the tangent line to the graph of $f(x)$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 $f(x) = mx + b$f(x)=mx+b, the derivative is the constant slope $m$m
- The derivative of a constant function is always zero, as the slope is horizontal
- Power Rule: For a function $f(x) = x^n$f(x)=xn, the derivative is $f'(x) = nx^{n-1}$f′(x)=nxn−1
- Example: If $f(x) = x^3$f(x)=x3, then $f'(x) = 3x^2$f′(x)=3x2

- Derivatives of common functions:
- $\frac{d}{dx} \sin(x) = \cos(x)$dxdsin(x)=cos(x)
- $\frac{d}{dx} \cos(x) = -\sin(x)$dxdcos(x)=−sin(x)
- $\frac{d}{dx} e^x = e^x$dxdex=ex
- $\frac{d}{dx} \ln(x) = \frac{1}{x}$dxdln(x)=x1

## Rules of the Game: Differentiation Techniques

- Sum Rule: The derivative of a sum is the sum of the derivatives
- $\frac{d}{dx} [f(x) + g(x)] = f'(x) + g'(x)$dxd[f(x)+g(x)]=f′(x)+g′(x)

- Difference Rule: The derivative of a difference is the difference of the derivatives
- $\frac{d}{dx} [f(x) - g(x)] = f'(x) - g'(x)$dxd[f(x)−g(x)]=f′(x)−g′(x)

- Constant Multiple Rule: Constants can be factored out when differentiating
- $\frac{d}{dx} [c \cdot f(x)] = c \cdot f'(x)$dxd[c⋅f(x)]=c⋅f′(x), where $c$c is a constant

- Product Rule: For two functions $f(x)$f(x) and $g(x)$g(x), the derivative of their product is:
- $\frac{d}{dx} [f(x) \cdot g(x)] = f(x) \cdot g'(x) + f'(x) \cdot g(x)$dxd[f(x)⋅g(x)]=f(x)⋅g′(x)+f′(x)⋅g(x)

- Quotient Rule: For two functions $f(x)$f(x) and $g(x)$g(x), the derivative of their quotient is:
- $\frac{d}{dx} [\frac{f(x)}{g(x)}] = \frac{g(x) \cdot f'(x) - f(x) \cdot g'(x)}{[g(x)]^2}$dxd[g(x)f(x)]=[g(x)]2g(x)⋅f′(x)−f(x)⋅g′(x)

- 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 $h(x) = f(g(x))$h(x)=f(g(x)), then $h'(x) = f'(g(x)) \cdot g'(x)$h′(x)=f′(g(x))⋅g′(x)
- Differentiate the outer function, then multiply by the derivative of the inner function
- Example: If $h(x) = \sin(x^2)$h(x)=sin(x2), then $h'(x) = \cos(x^2) \cdot 2x$h′(x)=cos(x2)⋅2x

- Implicit Differentiation: Used when a function is not explicitly defined as $y = f(x)$y=f(x)
- Differentiate both sides of the equation with respect to $x$x, treating $y$y as a function of $x$x
- Example: For the equation $x^2 + y^2 = 25$x2+y2=25, implicitly differentiating yields:
- $2x + 2y \cdot \frac{dy}{dx} = 0$2x+2y⋅dxdy=0
- Solve for $\frac{dy}{dx}$dxdy 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 $f'(x) > 0$f′(x)>0 on an interval, $f(x)$f(x) is increasing on that interval
- If $f'(x) < 0$f′(x)<0 on an interval, $f(x)$f(x) is decreasing on that interval

- Second Derivative Test: Determines the concavity of a function
- If $f''(x) > 0$f′′(x)>0 at a point, the graph is concave up at that point
- If $f''(x) < 0$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 $f''(x) = 0$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
- $f''(x) > 0$f′′(x)>0 indicates concave up, while $f''(x) < 0$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 $\lim_{x \to a} \frac{f(x)}{g(x)}$limx→ag(x)f(x) is an indeterminate form, then $\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}$limx→ag(x)f(x)=limx→ag′(x)f′(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 $x = f(t)$x=f(t) and $y = g(t)$y=g(t), then $\frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{g'(t)}{f'(t)}$dxdy=dx/dtdy/dt=f′(t)g′(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