∫Calculus I Unit 3 – Derivatives

What Are Derivatives?

• Derivatives measure the rate of change of a function at a particular point
• The derivative of a function $f(x)$ is denoted as $f'(x)$ and represents the instantaneous rate of change or slope of the tangent line at a specific point
• Derivatives can be used to determine the slope of a curve at any point, which helps analyze the behavior of functions
• The process of finding derivatives is called differentiation and involves applying specific rules and techniques
• Derivatives have numerous applications in various fields such as physics (velocity and acceleration), economics (marginal cost and revenue), and engineering (optimization problems)

Key Concepts and Terminology

• Limit is a fundamental concept in calculus that describes the value a function approaches as the input approaches a certain value
• Continuity refers to a function that has no breaks or gaps in its graph, meaning the function is defined at every point within its domain
• Differentiability is a property of a function that indicates whether it has a well-defined derivative at every point in its domain
• Tangent line is a straight line that touches a curve at a single point and has the same slope as the curve at that point
• Secant line is a line that intersects a curve at two points and is used to approximate the slope of the curve between those points
• Leibniz notation $\frac{dy}{dx}$ represents the derivative of $y$ with respect to $x$ and is an alternative to the prime notation $f'(x)$

Rules for Differentiation

• The power rule states that for a function $f(x) = x^n$, its derivative is $f'(x) = nx^{n-1}$
  • For example, if $f(x) = x^3$, then $f'(x) = 3x^2$
• The constant rule states that the derivative of a constant function is always 0
  • For example, if $f(x) = 5$, then $f'(x) = 0$
• The constant multiple rule states that if $f(x)$ is a function and $c$ is a constant, then the derivative of $c \cdot f(x)$ is $c \cdot f'(x)$
• The sum rule states that the derivative of a sum of functions is the sum of their derivatives
  • If $f(x) = g(x) + h(x)$, then $f'(x) = g'(x) + h'(x)$
• The product rule states that for two functions $u(x)$ and $v(x)$, the derivative of their product is $\frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)$
• The quotient rule states that for two functions $u(x)$ and $v(x)$, the derivative of their quotient is $\frac{d}{dx}[\frac{u(x)}{v(x)}] = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2}$

Applications of Derivatives

• Optimization problems involve finding the maximum or minimum values of a function within given constraints
  • Derivatives help determine the critical points where the function reaches its extrema
• Rate of change problems involve determining how quickly a quantity is changing at a specific point
  • Derivatives provide the instantaneous rate of change at any point on a function
• Related rates problems involve finding the rate at which one quantity is changing with respect to another related quantity
  • Derivatives allow us to establish relationships between the rates of change of different variables
• Approximation problems use derivatives to estimate the value of a function near a known point using linear approximation
• Marginal analysis in economics uses derivatives to determine the change in cost, revenue, or profit resulting from a small change in production or consumption

Common Derivative Functions

• The derivative of $f(x) = \sin(x)$ is $f'(x) = \cos(x)$
• The derivative of $f(x) = \cos(x)$ is $f'(x) = -\sin(x)$
• The derivative of $f(x) = e^x$ is $f'(x) = e^x$
  • This property makes the exponential function $e^x$ unique and important in calculus
• The derivative of $f(x) = \ln(x)$ is $f'(x) = \frac{1}{x}$
• The derivative of $f(x) = \tan(x)$ is $f'(x) = \sec^2(x)$
• The derivative of $f(x) = \cot(x)$ is $f'(x) = -\csc^2(x)$
• The derivative of $f(x) = \sec(x)$ is $f'(x) = \sec(x)\tan(x)$
• The derivative of $f(x) = \csc(x)$ is $f'(x) = -\csc(x)\cot(x)$

Graphing and Interpretation

• The sign of the derivative indicates whether the function is increasing (positive derivative) or decreasing (negative derivative) at a given point
• A derivative of zero at a point means the tangent line is horizontal, indicating a potential local maximum, local minimum, or inflection point
• The second derivative $f''(x)$ provides information about the concavity of the function
  • If $f''(x) > 0$, the function is concave up
  • If $f''(x) < 0$, the function is concave down
• Inflection points occur where the concavity of the function changes, and the second derivative is zero or undefined at these points
• Analyzing the first and second derivatives together helps sketch the general shape and behavior of a function

Problem-Solving Strategies

• Identify the type of problem (optimization, related rates, etc.) and the given information
• Determine the function or relationship that needs to be differentiated
• Apply the appropriate differentiation rules and techniques to find the derivative
• Set the derivative equal to zero to find critical points for optimization problems
• Use the derivative to answer the specific question asked in the problem
• Interpret the results in the context of the problem and check if the solution makes sense
• Sketch a graph of the function, if helpful, to visualize the problem and solution

Real-World Examples

• In physics, derivatives are used to calculate velocity (rate of change of position) and acceleration (rate of change of velocity)
  • For example, if an object's position is given by $s(t) = t^2 + 2t$, its velocity at any time $t$ is $v(t) = s'(t) = 2t + 2$
• In economics, derivatives are used to determine marginal cost (the change in total cost from producing one additional unit) and marginal revenue (the change in total revenue from selling one additional unit)
  • If the total cost function is $C(x) = 100 + 5x + 0.2x^2$, the marginal cost at a production level of $x$ units is $C'(x) = 5 + 0.4x$
• In engineering, derivatives are used to optimize the design of structures, machines, and systems
  • For instance, finding the dimensions of a cylindrical can that minimizes the surface area for a given volume involves setting up a function and finding its minimum using derivatives
• In finance, derivatives (such as options and futures contracts) are financial instruments whose value is based on the rate of change of an underlying asset or variable
  • The Black-Scholes model, which uses partial derivatives, is a widely used formula for pricing options contracts