Derivatives - Higher Order Differentiation
Recap: Derivatives measure the rate of change of a function
Higher order derivatives measure the rate of change of derivatives
Notation: f’(x) represents the first derivative of f(x)
Notation: f’’(x) represents the second derivative of f(x)
Higher order derivatives are denoted as f^n(x), where n is a positive integer
Finding Higher Order Derivatives
To find the nth derivative of a function, we differentiate the function n times
Example: Find the second derivative of f(x) = x^3
Step 1: Find the first derivative: f’(x) = 3x^2
Step 2: Find the second derivative: f’’(x) = 6x
Generally, to find the nth derivative of f(x), we differentiate f(x) n times
Rules for Differentiating Higher Order Derivatives
Rule 1: Constant multiple rule
If g(x) is a constant, then (g*f)^n(x) = g(f^n(x))
Rule 2: Sum/Difference rule
If f(x) and g(x) are functions, then (f + g)^n(x) = f^n(x) + g^n(x)
This rule also applies to subtraction
Rules for Differentiating Higher Order Derivatives (contd.)
Rule 3: Product rule
If f(x) and g(x) are functions, then (f*g)^n(x) = f^n(x) * g(x) + f(x) * g^n(x)
Rule 4: Quotient rule
If f(x) and g(x) are functions, then (f/g)^n(x) = [(f^n(x))*g(x) - f(x)*g^n(x)] / [g(x)]^2
Example 1: Finding Higher Order Derivatives
Find the third derivative of f(x) = sin(x)
Step 1: Find the first derivative: f’(x) = cos(x)
Step 2: Find the second derivative: f’’(x) = -sin(x)
Step 3: Find the third derivative: f’’’(x) = -cos(x)
In general, differentiating sin(x) n times will give us cos(x) or sin(x) with a negative sign depending on the value of n
Example 2: Finding Higher Order Derivatives
Find the fourth derivative of f(x) = e^x
Step 1: Find the first derivative: f’(x) = e^x
Step 2: Find the second derivative: f’’(x) = e^x
Step 3: Find the third derivative: f’’’(x) = e^x
Step 4: Find the fourth derivative: f’’’’(x) = e^x
In general, differentiating e^x n times will always give us e^x
Example 3: Finding Higher Order Derivatives
Find the fourth derivative of f(x) = x^4 + 2x^3 - 4
Step 1: Find the first derivative: f’(x) = 4x^3 + 6x^2
Step 2: Find the second derivative: f’’(x) = 12x^2 + 12x
Step 3: Find the third derivative: f’’’(x) = 24x + 12
Step 4: Find the fourth derivative: f’’’’(x) = 24
In general, differentiating any polynomial function n times will give us a constant term if n is greater than the degree of the polynomial
Summary
Higher order derivatives measure the rate of change of derivatives
The nth derivative of a function is obtained by differentiating the function n times
Rules for differentiating higher order derivatives: constant multiple rule, sum/difference rule, product rule, quotient rule
Examples: sin(x), e^x, polynomial functions
Derivatives - Higher Order Differentiation
Recap: Derivatives measure the rate of change of a function
Higher order derivatives measure the rate of change of derivatives
Notation: f’(x) represents the first derivative of f(x)
Notation: f’’(x) represents the second derivative of f(x)
Higher order derivatives are denoted as f^n(x), where n is a positive integer
Finding Higher Order Derivatives
To find the nth derivative of a function, we differentiate the function n times
Example: Find the second derivative of f(x) = x^3
Step 1: Find the first derivative: f’(x) = 3x^2
Step 2: Find the second derivative: f’’(x) = 6x
Generally, to find the nth derivative of f(x), we differentiate f(x) n times
Rules for Differentiating Higher Order Derivatives
Rule 1: Constant multiple rule
If g(x) is a constant, then (g*f)^n(x) = g(f^n(x))
Rule 2: Sum/Difference rule
If f(x) and g(x) are functions, then (f + g)^n(x) = f^n(x) + g^n(x)
This rule also applies to subtraction
Rule 3: Product rule
If f(x) and g(x) are functions, then (f*g)^n(x) = f^n(x) * g(x) + f(x) * g^n(x)
Rules for Differentiating Higher Order Derivatives (contd.)
Rule 4: Quotient rule
If f(x) and g(x) are functions, then (f/g)^n(x) = [(f^n(x))*g(x) - f(x)*g^n(x)] / [g(x)]^2
Rule 5: Chain rule
If y = f(u) and u = g(x), then dy/dx = (dy/du) * (du/dx)
Rule 6: Power rule
If y = x^n, then dy/dx = nx^(n-1)
Example 1: Finding Higher Order Derivatives
Find the third derivative of f(x) = sin(x)
Step 1: Find the first derivative: f’(x) = cos(x)
Step 2: Find the second derivative: f’’(x) = -sin(x)
Step 3: Find the third derivative: f’’’(x) = -cos(x)
Example 2: Finding Higher Order Derivatives
Find the fourth derivative of f(x) = e^x
Step 1: Find the first derivative: f’(x) = e^x
Step 2: Find the second derivative: f’’(x) = e^x
Step 3: Find the third derivative: f’’’(x) = e^x
Step 4: Find the fourth derivative: f’’’’(x) = e^x
Example 3: Finding Higher Order Derivatives
Find the fourth derivative of f(x) = x^4 + 2x^3 - 4
Step 1: Find the first derivative: f’(x) = 4x^3 + 6x^2
Step 2: Find the second derivative: f’’(x) = 12x^2 + 12x
Step 3: Find the third derivative: f’’’(x) = 24x + 12
Step 4: Find the fourth derivative: f’’’’(x) = 24
Example 4: Finding Higher Order Derivatives
Find the second derivative of f(x) = 5x^2 - 3x + 2
Step 1: Find the first derivative: f’(x) = 10x - 3
Step 2: Find the second derivative: f’’(x) = 10
In general, differentiating any quadratic function twice will give us a constant term
Example 5: Finding Higher Order Derivatives
Find the third derivative of f(x) = (2x^3 + 3x^2 - x + 1) / (x^2 + 1)
Step 1: Find the first derivative: f’(x) = (6x^2 + 6x - 1) / (x^2 + 1) - (2x^3 + 3x^2 - x + 1)(2x) / (x^2 + 1)^2
Step 2: Find the second derivative: f’’(x) = (12x + 6) / (x^2 + 1) - (6x^2 + 6x - 1)(2x) / (x^2 + 1)^2
Step 3: Find the third derivative: f’’’(x) = (12 - 12x^2)(x^2 + 1)^2 - (6x^2 + 6x - 1)(4x)(x^2 + 1) / (x^2 + 1)^4
The third derivative of the given function is quite complex, involving both polynomial and rational expressions
Example 6: Finding Higher Order Derivatives
Find the third derivative of f(x) = ln(x^2 - 4)
Step 1: Find the first derivative: f’(x) = (2x) / (x^2 - 4)
Step 2: Find the second derivative: f’’(x) = (2(x^2 - 4) - 2x(2x)) / (x^2 - 4)^2
Step 3: Find the third derivative: f’’’(x) = [2((x^2 - 4)^2) - 2x(2(x^2 - 4)) - 2(x^2 - 4)(4x)] / (x^2 - 4)^3
The third derivative of the given function involves complex algebraic manipulations with logarithmic functions
Applications of Higher Order Derivatives
Higher order derivatives can be used to analyze the shape and behavior of functions
They can be used to find the concavity and points of inflection of a function
They can help us identify the maximum and minimum points
Higher order derivatives are also important in physics, engineering, and economics for modeling and optimization problems
Second Derivative Test
The second derivative of a function can be used to determine its concavity and locate points of inflection
Second derivative test: If the second derivative f’’(x) is positive, the function is concave up. If f’’(x) is negative, the function is concave down
Points of inflection occur where the sign of the second derivative changes from positive to negative or vice versa
Second Derivative Test (contd.)
Example: Let f(x) = x^3 - 6x^2 + 9x + 1
Step 1: Find the first derivative: f’(x) = 3x^2 - 12x + 9
Step 2: Find the second derivative: f’’(x) = 6x - 12
The second derivative f’’(x) is positive for x > 2 and negative for x < 2
Second Derivative Test (contd.)
Example: Let f(x) = x^3 - 6x^2 + 9x + 1 (contd.)
The function is concave up for x > 2 and concave down for x < 2
We can conclude that the point of inflection occurs at x = 2
Finding Maximum and Minimum Points
Higher order derivatives can help us find the maximum and minimum points of a function
The first derivative gives us critical points where the derivative is zero or undefined
The second derivative tells us whether each critical point is a maximum or minimum
If f’’(x) > 0 at a critical point c, the function has a local minimum at c
If f’’(x) < 0 at a critical point c, the function has a local maximum at c
Example: Finding Maximum and Minimum Points
Let f(x) = x^3 - 6x^2 + 9x + 1
Step 1: Find the first derivative: f’(x) = 3x^2 - 12x + 9
Step 2: Find the second derivative: f’’(x) = 6x - 12
Critical points occur when f’(x) = 0 or f’(x) is undefined
Example: Finding Maximum and Minimum Points (contd.)
Let f(x) = x^3 - 6x^2 + 9x + 1 (contd.)
Setting f’(x) = 0, we get 3x^2 - 12x + 9 = 0
Solving for x, we find x = 1 and x = 3 as critical points
Evaluating f’’(x) at these critical points, we find that f’’(1) = -6 and f’’(3) = 6
We have a local maximum at x = 1 and a local minimum at x = 3
Summary
Higher order derivatives measure the rate of change of derivatives
The nth derivative of a function is obtained by differentiating the function n times
Rules for differentiating higher order derivatives: constant multiple rule, sum/difference rule, product rule, quotient rule, chain rule, power rule
Examples: sin(x), e^x, polynomial functions, logarithmic functions
Higher order derivatives can be used to analyze the shape, concavity, and maximum/minimum points of a function