Derivatives - Examples of Product rule

• The product rule is a method used to find the derivative of a product of two functions.
• It is given by the formula: (fg)’ = f’g + fg'
• Let’s look at some examples to understand how to apply the product rule.

Example 1: Find the derivative of f(x) = x² * sin(x)

• Let’s apply the product rule to find the derivative of the given function.
• Step 1: Identify the functions f(x) and g(x).
  • f(x) = x²
  • g(x) = sin(x)
• Step 2: Find the derivatives of f(x) and g(x).
  • f’(x) = 2x
  • g’(x) = cos(x)
• Step 3: Apply the product rule formula.
  • (f * g)’ = f’g + fg'
  • (x² * sin(x))’ = (2x * sin(x)) + (x² * cos(x))
• We can simplify further if needed.

Example 2: Find the derivative of f(x) = (1 + 3x) * e^x

• Let’s apply the product rule to find the derivative of the given function.
• Step 1: Identify the functions f(x) and g(x).
  • f(x) = 1 + 3x
  • g(x) = e^x
• Step 2: Find the derivatives of f(x) and g(x).
  • f’(x) = 3
  • g’(x) = e^x
• Step 3: Apply the product rule formula.
  • (f * g)’ = f’g + fg'
  • ((1 + 3x) * e^x)’ = (3 * e^x) + ((1 + 3x) * e^x)
• We can simplify further if needed.

Derivatives - Examples of Quotient rule

• The quotient rule is a method used to find the derivative of a quotient of two functions.
• It is given by the formula: (f/g)’ = (f’g - fg’) / g²
• Let’s look at some examples to understand how to apply the quotient rule.

Example 1: Find the derivative of f(x) = (2x + 1) / x

• Let’s apply the quotient rule to find the derivative of the given function.
• Step 1: Identify the functions f(x) and g(x).
  • f(x) = 2x + 1
  • g(x) = x
• Step 2: Find the derivatives of f(x) and g(x).
  • f’(x) = 2
  • g’(x) = 1
• Step 3: Apply the quotient rule formula.
  • (f/g)’ = (f’g - fg’) / g²
  • ((2x + 1) / x)’ = ((2 * x - (2x + 1)) / x²)
• We can simplify further if needed.

Example 2: Find the derivative of f(x) = (3x² - 2) / (x + 1)

• Let’s apply the quotient rule to find the derivative of the given function.
• Step 1: Identify the functions f(x) and g(x).
  • f(x) = 3x² - 2
  • g(x) = x + 1
• Step 2: Find the derivatives of f(x) and g(x).
  • f’(x) = 6x
  • g’(x) = 1
• Step 3: Apply the quotient rule formula.
  • (f/g)’ = (f’g - fg’) / g²
  • ((3x² - 2) / (x + 1))’ = (((6x(x + 1)) - ((3x² - 2) * 1)) / (x + 1)²)
• We can simplify further if needed.

Derivatives - Examples of Chain rule

• The chain rule is a method used to find the derivative of a composition of two or more functions.
• It is given by the formula: (f(g(x)))’ = f’(g(x)) * g’(x)
• Let’s look at some examples to understand how to apply the chain rule.

Example 1: Find the derivative of f(x) = sin(3x²)

• Let’s apply the chain rule to find the derivative of the given function.
• Step 1: Identify the functions f(x) and g(x).
  • f(x) = sin(x)
  • g(x) = 3x²
• Step 2: Find the derivatives of f(x) and g(x).
  • f’(x) = cos(x)
  • g’(x) = 6x
• Step 3: Apply the chain rule formula.
  • (f(g(x)))’ = f’(g(x)) * g’(x)
  • (sin(3x²))’ = cos(3x²) * 6x
• We can simplify further if needed.

Example 2: Find the derivative of f(x) = e^(2x + 1)

• Let’s apply the chain rule to find the derivative of the given function.
• Step 1: Identify the functions f(x) and g(x).
  • f(x) = e^x
  • g(x) = 2x + 1
• Step 2: Find the derivatives of f(x) and g(x).
  • f’(x) = e^x
  • g’(x) = 2
• Step 3: Apply the chain rule formula.
  • (f(g(x)))’ = f’(g(x)) * g’(x)
  • (e^(2x + 1))’ = e^(2x + 1) * 2
• We can simplify further if needed.

Derivatives - Summary

• The product rule is used to find the derivative of a product of two functions.
• The quotient rule is used to find the derivative of a quotient of two functions.
• The chain rule is used to find the derivative of a composition of two or more functions.
• These rules are essential tools for finding derivatives in calculus.
• Practice applying these rules to various functions and equations to strengthen your understanding.

Derivatives - Examples of Product Rule

• Example 1:
  • f(x) = x² * sin(x)
  • Apply product rule: (x² * sin(x))’ = (2x * sin(x)) + (x² * cos(x))
• Example 2:
  • f(x) = (1 + 3x) * e^x
  • Apply product rule: ((1 + 3x) * e^x)’ = (3 * e^x) + ((1 + 3x) * e^x)

Derivatives - Examples of Quotient Rule

• Example 1:
  • f(x) = (2x + 1) / x
  • Apply quotient rule: ((2x + 1) / x)’ = ((2 * x - (2x + 1)) / x²)
• Example 2:
  • f(x) = (3x² - 2) / (x + 1)
  • Apply quotient rule: (((6x(x + 1)) - ((3x² - 2) * 1)) / (x + 1)²)

Derivatives - Examples of Chain Rule

• Example 1:
  • f(x) = sin(3x²)
  • Apply chain rule: (sin(3x²))’ = cos(3x²) * 6x
• Example 2:
  • f(x) = e^(2x + 1)
  • Apply chain rule: (e^(2x + 1))’ = e^(2x + 1) * 2

Optimization Problems - Intro

• Optimization problems involve finding the maximum or minimum values of a function.
• These problems can be solved using calculus techniques such as derivatives.
• There are two main steps in solving optimization problems:
  1. Setting up the function that represents the quantity to be optimized.
  2. Finding the derivative and solving for critical points.

Optimization Problems - Example 1

• Problem: Find the dimensions of a rectangle with maximum area that has a fixed perimeter of 20 units.
• Step 1: Set up the function.
  • Let the length and width of the rectangle be l and w respectively.
  • The perimeter is given by P = 2l + 2w.
  • The area is given by A = lw.
• Step 2: Find the derivative and critical points.
  • Derivative of A: A’ = w - (20 - 2w)/2 = w - 10 + w = 2w - 10.
  • Solve A’ = 0 to find critical points.
• Step 3: Evaluate critical points and find the maximum.

Optimization Problems - Example 2

• Problem: Find the dimensions of a rectangular box with maximum volume that has a surface area of 100 square units.
• Step 1: Set up the function.
  • Let the length, width, and height of the box be l, w, and h respectively.
  • The surface area is given by SA = 2lw + 2lh + 2wh.
  • The volume is given by V = lwh.
• Step 2: Find the derivative and critical points.
  • Derivative of V: V’ = wh + lw + lh = 100 / (2h + 2l + 2w).
  • Solve V’ = 0 to find critical points.
• Step 3: Evaluate critical points and find the maximum.

Integration - Introduction

• Integration is the reverse process of differentiation.
• It involves finding the antiderivative (also called primitive) of a function.
• An indefinite integral in the form ∫ f(x) dx represents the antiderivative of f(x).
• Definite integration is used to find the area under a curve between two points.
• The definite integral is represented by the notation ∫[a, b] f(x) dx.

Integration - Basic Rules

• Rule 1: Power Rule
  • ∫ x^n dx = (x^(n+1)) / (n+1) + C, where n is any real number except -1.
• Rule 2: Constant Multiple Rule
  • ∫ k * f(x) dx = k * ∫ f(x) dx, where k is a constant.
• Rule 3: Sum Rule
  • ∫ (f(x) + g(x)) dx = ∫ f(x) dx + ∫ g(x) dx.

Integration - Example 1

• Example: Find ∫(3x² + 2x - 1) dx
• Solution:
  • Apply the sum rule: ∫(3x² + 2x - 1) dx = ∫(3x²) dx + ∫(2x) dx - ∫(1) dx.
  • Apply the power rule: ∫(3x²) dx = (3/3)x³ + C₁ = x³ + C₁.
  • Apply the power rule: ∫(2x) dx = (2/2)x² + C₂ = x² + C₂.
  • ∫(1) dx = x + C₃.
  • Combine the results: ∫(3x² + 2x - 1) dx = x³ + x² + x + C.

Integration - Example 2

• Example: Find ∫(5sin(x) + 3cos(x)) dx
• Solution:
  • Apply the sum rule: ∫(5sin(x) + 3cos(x)) dx = ∫(5sin(x)) dx + ∫(3cos(x)) dx.
  • Apply the antiderivative of sin(x): ∫(5sin(x)) dx = -5cos(x) + C₁.
  • Apply the antiderivative of cos(x): ∫(3cos(x)) dx = 3sin(x) + C₂.
  • Combine the results: ∫(5sin(x) + 3cos(x)) dx = -5cos(x) + 3sin(x) + C.

Derivatives - Examples of Product rule

• Example 1:
  • f(x) = x² * sin(x)
  • Apply product rule: (x² * sin(x))’ = (2x * sin(x)) + (x² * cos(x))
• Example 2:
  • f(x) = (1 + 3x) * e^x
  • Apply product rule: ((1 + 3x) * e^x)’ = (3 * e^x) + ((1 + 3x) * e^x)

Derivatives - Examples of Quotient rule

• Example 1:
  • f(x) = (2x + 1) / x
  • Apply quotient rule: ((2x + 1) / x)’ = ((2 * x - (2x + 1)) / x²)
• Example 2:
  • f(x) = (3x² - 2) / (x + 1)
  • Apply quotient rule: (((6x(x + 1)) - ((3x² - 2) * 1)) / (x + 1)²)

Derivatives - Examples of Chain rule

• Example 1:
  • f(x) = sin(3x²)
  • Apply chain rule: (sin(3x²))’ = cos(3x²) * 6x
• Example 2:
  • f(x) = e^(2x + 1)
  • Apply chain rule: (e^(2x + 1))’ = e^(2x + 1) * 2

Optimization Problems - Intro

• Optimization problems involve finding the maximum or minimum values of a function.
• These problems can be solved using calculus techniques such as derivatives.
• There are two main steps in solving optimization problems:
  1. Setting up the function that represents the quantity to be optimized.
  2. Finding the derivative and solving for critical points.

Optimization Problems - Example 1

• Problem: Find the dimensions of a rectangle with maximum area that has a fixed perimeter of 20 units.
• Step 1: Set up the function.
  • Let the length and width of the rectangle be l and w respectively.
  • The perimeter is given by P = 2l + 2w.
  • The area is given by A = lw.
• Step 2: Find the derivative and critical points.
  • Derivative of A: A’ = w - (20 - 2w)/2 = w - 10 + w = 2w - 10.
  • Solve A’ = 0 to find critical points.
• Step 3: Evaluate critical points and find the maximum.

Optimization Problems - Example 2

• Problem: Find the dimensions of a rectangular box with maximum volume that has a surface area of 100 square units.
• Step 1: Set up the function.
  • Let the length, width, and height of the box be l, w, and h respectively.
  • The surface area is given by SA = 2lw + 2lh + 2wh.
  • The volume is given by V = lwh.
• Step 2: Find the derivative and critical points.
  • Derivative of V: V’ = wh + lw + lh = 100 / (2h + 2l + 2w).
  • Solve V’ = 0 to find critical points.
• Step 3: Evaluate critical points and find the maximum.

Integration - Introduction

• Integration is the reverse process of differentiation.
• It involves finding the antiderivative (also called primitive) of a function.
• An indefinite integral in the form ∫ f(x) dx represents the antiderivative of f(x).
• Definite integration is used to find the area under a curve between two points.
• The definite integral is represented by the notation ∫[a, b] f(x) dx.

Integration - Basic Rules

• Rule 1: Power Rule
  • ∫ x^n dx = (x^(n+1)) / (n+1) + C, where n is any real number except -1.
• Rule 2: Constant Multiple Rule
  • ∫ k * f(x) dx = k * ∫ f(x) dx, where k is a constant.
• Rule 3: Sum Rule
  • ∫ (f(x) + g(x)) dx = ∫ f(x) dx + ∫ g(x) dx.

Integration - Example 1

• Example: Find ∫(3x² + 2x - 1) dx
• Solution:
  • Apply the sum rule: ∫(3x² + 2x - 1) dx = ∫(3x²) dx + ∫(2x) dx - ∫(1) dx.
  • Apply the power rule: ∫(3x²) dx = (3/3)x³ + C₁ = x³ + C₁.
  • Apply the power rule: ∫(2x) dx = (2/2)x² + C₂ = x² + C₂.
  • ∫(1) dx = x + C₃.
  • Combine the results: ∫(3x² + 2x - 1) dx = x³ + x² + x + C.

Integration - Example 2

• Example: Find ∫(5sin(x) +