Slide 1: Introduction to Derivatives

  • Definition: The derivative of a function represents the rate at which the function value changes with respect to the independent variable.
  • Denoted by f’(x), dy/dx, or d/dx(f(x)).
  • It measures the slope of the curve at a particular point on the graph.
  • Applications of derivatives in various fields such as physics, economics, and engineering.
  • The derivative helps in determining maximum and minimum values of a function.

Slide 2: Basic Notations for Derivatives

  • Notation for derivative: f’(x) or dy/dx.
  • Notation for second derivative: f’’(x) or d^2y/dx^2.
  • Higher order derivatives: f’’’, f’’’’ and so on.
  • The derivative at a specific point is written as f’(a) or dy/dx|ₓ=a.
  • Differentiation of a function can be represented as f(x) → f’(x) or f(x) → df(x)/dx.

Slide 3: Differentiation Rules

  • Power rule: For any constant n and a function f(x), (d/dx)(x^n) = nx^(n-1).
  • Constant rule: For any constant c, (d/dx)(c) = 0.
  • Sum rule: For two functions f(x) and g(x), (d/dx)(f(x) + g(x)) = f’(x) + g’(x).
  • Product rule: For two functions f(x) and g(x), (d/dx)(f(x) * g(x)) = f’(x) * g(x) + f(x) * g’(x).
  • Quotient rule: For two functions f(x) and g(x), (d/dx)(f(x) / g(x)) = (f’(x) * g(x) - f(x) * g’(x)) / g(x)^2.

Slide 4: Derivatives of Trigonometric Functions

  • Derivative of sin(x) = cos(x).
  • Derivative of cos(x) = -sin(x).
  • Derivative of tan(x) = sec^2(x).
  • Derivative of cot(x) = -cosec^2(x).
  • Derivative of sec(x) = sec(x) * tan(x).
  • Derivative of cosec(x) = -cosec(x) * cot(x).

Slide 5: Derivatives of Exponential Functions

  • Derivative of e^x = e^x.
  • Derivative of a^x (where a is a constant) = ln(a) * a^x.
  • Derivative of ln(x) = 1/x.
  • Derivative of log_a(x) = 1/(x * ln(a)).

Slide 6: Derivatives of Logarithmic Functions

  • Derivative of ln(x) = 1/x.
  • Derivative of log_a(x) = 1/(x * ln(a)).

Slide 7: Derivatives of Inverse Trigonometric Functions

  • Derivative of sin^(-1)(x) = 1/√(1 - x^2).
  • Derivative of cos^(-1)(x) = -1/√(1 - x^2).
  • Derivative of tan^(-1)(x) = 1/(1 + x^2).
  • Derivative of cot^(-1)(x) = -1/(1 + x^2).
  • Derivative of sec^(-1)(x) = 1/(x√(x^2 - 1)).
  • Derivative of cosec^(-1)(x) = -1/(x√(x^2 - 1)).

Slide 8: Differentiation of Composite Functions

  • Chain rule: If y = f(g(x)), then dy/dx = dy/dg * dg/dx.
  • Example: Differentiate y = sin(2x^2). Let u = 2x^2, v = sin(u).
    • dy/dx = dy/du * du/dx = cos(u) * 4x = 4x * cos(2x^2).

Slide 9: Differentiation of Implicit Functions

  • Implicit differentiation: Differentiating both sides of an equation containing both dependent and independent variables.
  • Steps for implicit differentiation:
    1. Differentiate both sides of the equation.
    2. Treat y as a function of x and use chain rule for terms containing y.
    3. Solve the resulting equation for dy/dx.
  • Example: Differentiate x^2 + y^2 = 4xy.
    • Differentiating both sides: 2x + 2y * dy/dx = 4x * dy/dx + 4y.
    • Rearranging and solving for dy/dx: dy/dx = (2x - 2y)/(2y - 4x).

Slide 10: Derivatives and Rates of Change

  • Derivatives represent rates of change.
  • In economics, derivatives can be used to analyze rates of change of certain quantities.
  • Example: Marginal cost represents the rate of change in total cost with respect to the total quantity produced.
  • Example: Marginal revenue represents the rate of change in total revenue with respect to the total quantity sold.
  • Understanding derivatives helps in making decisions about production levels and pricing strategies.

Derivatives - Examples of rate of change in Economics

  • Marginal cost: The rate of change in total cost with respect to the quantity produced.
    • Mathematically represented as MC = dC/dQ, where MC is the marginal cost, C is the total cost, and Q is the quantity produced.
    • Example: If the total cost function is given by C = 0.2Q^2 + 50Q + 100, then the marginal cost can be calculated by differentiating the total cost function.
  • Marginal revenue: The rate of change in total revenue with respect to the quantity sold.
    • Mathematically represented as MR = dR/dQ, where MR is the marginal revenue, R is the total revenue, and Q is the quantity sold.
    • Example: If the total revenue function is given by R = 50Q - 0.5Q^2, then the marginal revenue can be calculated by differentiating the total revenue function.
  • Profit maximization: In economics, derivatives are used to find the quantity at which the profit is maximized.
    • By setting the marginal cost equal to the marginal revenue, we can find the optimal quantity.
    • Example: If the marginal cost and marginal revenue functions are given, we can equate them and solve for the quantity.
  • Price elasticity of demand: The rate at which quantity demanded changes with respect to price changes.
    • Mathematically represented as PED = (dQ/dP) * (P/Q), where PED is the price elasticity of demand, Q is the quantity demanded, and P is the price.
    • Example: If the demand function is given by Q = 100 - 2P, then the price elasticity of demand can be calculated using derivatives.
  • Cost optimization: Derivatives can be used to find the minimum cost for a given production level.
    • By minimizing the cost function using derivatives, we can optimize the production level.
    • Example: If the cost function is given by C = 0.1Q^3 + 50Q^2 + 5000, we can differentiate it to find the minimum cost.

Related Rates - Applications in Physics and Engineering

  • Related rates problems involve finding the rate of change of one quantity with respect to another.
  • These problems often arise in physics and engineering, where multiple variables are related by an equation.
  • Steps to solve related rates problems:
    1. Identify the variables and their rates of change.
    2. Formulate an equation that relates the variables.
    3. Differentiate the equation with respect to time.
    4. Substitute the given values and solve for the desired rate.
  • Example: A ladder is sliding down a wall. Find the rate at which the ladder height is decreasing when the base is moving away at a certain rate.
  • Example: A cylindrical tank is being filled with water. Find the rate at which the water level is rising when the volume is increasing at a certain rate.
  • Example: A balloon is being inflated. Find the rate at which the radius is increasing when the volume is increasing at a certain rate.
  • Example: Two cars are approaching an intersection, moving at different speeds. Find the rate at which the distance between them is changing.
  • Example: A baseball is thrown upwards. Find the rate at which the height is changing at an instant when the velocity is known.

Higher Order Derivatives

  • Higher order derivatives represent the rate at which the rate of change is changing.
  • Second derivative: The derivative of the first derivative.
    • Mathematically represented as f’’(x) or d^2f(x)/dx^2.
    • It measures the rate at which the slope of the function is changing.
    • Interpretation: Positive second derivative indicates concavity upwards, negative second derivative indicates concavity downwards, and zero second derivative indicates an inflection point.
    • Example: If f’(x) = 3x^2, then f’’(x) = 6x.
  • Third derivative: The derivative of the second derivative.
    • Mathematically represented as f’’’(x) or d^3f(x)/dx^3.
    • It measures the rate at which the rate of change of the function is changing.
    • Example: If f’’(x) = 6x, then f’’’(x) = 6.
  • Fourth derivative: The derivative of the third derivative.
    • Mathematically represented as f’’’’(x) or d^4f(x)/dx^4.
    • Example: If f’’’(x) = 6, then f’’’’(x) = 0.
  • Higher order derivatives can be used to analyze the behavior of functions in various disciplines such as physics, economics, and engineering.

Applications of Derivatives - Optimization

  • Optimization problems involve finding maximum or minimum values of a function.
  • Derivatives can be used to solve optimization problems by evaluating critical points.
  • Steps to solve optimization problems using derivatives:
    1. Find the derivative of the function representing the quantity to be optimized.
    2. Set the derivative equal to zero and solve for the independent variable to find critical points.
    3. Evaluate each critical point to determine if it is a maximum or minimum.
    4. Verify the obtained maximum or minimum by checking the endpoints or using the second derivative test.
  • Example: A farmer has 200 meters of fencing and wants to enclose a rectangular field. Find the dimensions that maximize the area.
  • Example: A company manufactures a cylindrical can with a fixed volume. Find the dimensions that minimize the surface area of the can.
  • Example: A wire is to be cut into two pieces. One piece will be bent into a square and the other into a circle. Find the lengths of the pieces to maximize the total area.

Applications of Derivatives - Curve Sketching

  • Curve sketching involves analyzing the properties of a function graphically.
  • Derivatives can be used to determine the behavior of a function and sketch its graph.
  • Steps for curve sketching:
    1. Find the domain and range of the function.
    2. Determine the x and y intercepts by setting the function equal to zero.
    3. Find the critical points by setting the derivative equal to zero.
    4. Determine the concavity by evaluating the second derivative.
    5. Find the asymptotes, if any, by evaluating the limit of the function.
    6. Sketch the graph using the information obtained.
  • Example: Sketch the graph of f(x) = x^3 - 3x^2 - 9x + 10.
  • Example: Sketch the graph of f(x) = e^x * sin(x).
  • Example: Sketch the graph of f(x) = x^2 / (x^2 - 1).

Applications of Derivatives - Linear Approximation

  • Linear approximation is a method to estimate the value of a function near a known point.
  • The tangent line to the function at the known point is used as an approximation of the function.
  • Linear approximation formula: f(x) ≈ f(a) + f’(a)(x - a).
  • Example: Use linear approximation to estimate √8.
    • Take f(x) = √x, a = 9, and x = 8 in the formula.
    • Substitute the values and calculate the approximate value of √8.
  • Example: Use linear approximation to estimate sin(0.2π).
    • Take f(x) = sin(x), a = 0, and x = 0.2π in the formula.
    • Substitute the values and calculate the approximate value of sin(0.2π).
  • Linear approximation is useful in situations where precise calculations are not feasible or necessary.

Applications of Derivatives - Motion Problems

  • Motion problems involve finding the position, velocity, or acceleration of an object.
  • Derivatives can be used to analyze the motion of an object based on given information.
  • Steps to solve motion problems:
    1. Identify the variable to be determined (position, velocity, or acceleration).
    2. Set up an equation relating the variables based on the given information.
    3. Differentiate the equation with respect to time to find the derivative.
    4. Substitute the given values and solve for the desired variable.
  • Example: A particle moves along a straight line. Find its velocity if the position function is given.
  • Example: A car accelerates uniformly from rest. Find its position if the acceleration function is known.
  • Example: A projectile is launched vertically. Find its maximum height and total time of flight given the initial velocity.

Applications of Derivatives - Exponential Growth and Decay

  • Exponential growth models represent the growth of a quantity over time.
  • Exponential decay models represent the decay or decrease of a quantity over time.
  • Derivatives can be used to analyze exponential growth and decay.
  • Example: Find the growth rate of a population if the population follows an exponential growth model.
  • Example: Find the decay rate of a radioactive substance if the substance follows an exponential decay model.
  • The derivative of an exponential function is proportional to the value of the function itself.

Summary

  • Derivatives represent the rate at which a function changes.
  • They have various applications in different fields such as economics, physics, and engineering.
  • Differentiation rules can be used to find derivatives of different types of functions.
  • Higher-order derivatives represent the rates of change of rates of change.
  • Related rates problems involve finding the rates of change of two related variables.
  • Optimization problems involve finding maximum or minimum values of a function.
  • Curve sketching helps to analyze the properties of a function graphically.
  • Linear approximation is a method to estimate the value of a function near a known point.
  • Motion problems involve analyzing the position, velocity, and acceleration of objects.
  • Exponential growth and decay models represent the growth or decay of quantities over time.

Questions?

  • Please feel free to ask any questions you may have.
  • Let’s revise and practice some examples together.
  1. Derivatives - Examples of rate of change in Economics
  • Marginal cost: The rate of change in total cost with respect to the quantity produced.
    • Mathematically represented as MC = dC/dQ, where MC is the marginal cost, C is the total cost, and Q is the quantity produced.
    • Example: If the total cost function is given by C = 0.2Q^2 + 50Q + 100, then the marginal cost can be calculated by differentiating the total cost function.
  • Marginal revenue: The rate of change in total revenue with respect to the quantity sold.
    • Mathematically represented as MR = dR/dQ, where MR is the marginal revenue, R is the total revenue, and Q is the quantity sold.
    • Example: If the total revenue function is given by R = 50Q - 0.5Q^2, then the marginal revenue can be calculated by differentiating the total revenue function.
  • Profit maximization: In economics, derivatives are used to find the quantity at which the profit is maximized.
    • By setting the marginal cost equal to the marginal revenue, we can find the optimal quantity.
    • Example: If the marginal cost and marginal revenue functions are given, we can equate them and solve for the quantity.
  • Price elasticity of demand: The rate at which quantity demanded changes with respect to price changes.
    • Mathematically represented as PED = (dQ/dP) * (P/Q), where PED is the price elasticity of demand, Q is the quantity demanded, and P is the price.
    • Example: If the demand function is given by Q = 100 - 2P, then the price elasticity of demand can be calculated using derivatives.
  • Cost optimization: Derivatives can be used to find the minimum cost for a given production level.
    • By minimizing the cost function using derivatives, we can optimize the production level.
    • Example: If the cost function is given by C = 0.1Q^3 + 50Q^2 + 5000, we can differentiate it to find the minimum cost.
  1. Related Rates - Applications in Physics and Engineering
  • Related rates problems involve finding the rate of change of one quantity with respect to another.
  • These problems often arise in physics and engineering, where multiple variables are related by an equation.
  • Steps to solve related rates problems:
    1. Identify the variables and their rates of change.
    2. Formulate an equation that relates the variables.
    3. Differentiate the equation with respect to time.
    4. Substitute the given values and solve for the desired rate.
  • Example: A ladder is sliding down a wall. Find the rate at which the ladder height is decreasing when the base is moving away at a certain rate.
  • Example: A cylindrical tank is being filled with water. Find the rate at which the water level is rising when the volume is increasing at a certain rate.
  • Example: A balloon is being inflated. Find the rate at which the radius is increasing when the volume is increasing at a certain rate.
  • Example: Two cars are approaching an intersection, moving at different speeds. Find