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6.3 Higher-order derivatives

3 min read•july 22, 2024

Higher-order derivatives take differentiation to the next level. They're like Russian nesting dolls of math, where each derivative is nested inside the previous one. Understanding these can unlock deeper insights into how functions behave.

These derivatives are crucial in physics and engineering. They help us grasp complex motions, like and , and solve tricky problems. Mastering higher-order derivatives opens doors to advanced calculus applications.

Higher-Order Derivatives

Concept of higher-order derivatives

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Derivatives of derivatives
- obtained by differentiating the first derivative
- obtained by differentiating the second derivative and so on
Notation for higher-order derivatives
- $f'(x)$ denotes the first derivative of $f(x)$
- $[f''(x)](https://www.fiveableKeyTerm:f''(x))$ denotes the second derivative of $f(x)$ (derivative of the first derivative)
- $[f'''(x)](https://www.fiveableKeyTerm:f'''(x))$ denotes the third derivative of $f(x)$ (derivative of the second derivative)
- $[f^{(n)}(x)](https://www.fiveableKeyTerm:f^{(n)}(x))$ denotes the $n$ th derivative of $f(x)$ (derivative taken $n$ times)
for higher-order derivatives
- $\frac{d}{dx}f(x)$ represents the first derivative of $f(x)$
- $\frac{d^2}{dx^2}f(x)$ represents the second derivative of $f(x)$ (derivative of the first derivative)
- $\frac{d^3}{dx^3}f(x)$ represents the third derivative of $f(x)$ (derivative of the second derivative)
- $\frac{d^n}{dx^n}f(x)$ represents the $n$ th derivative of $f(x)$ (derivative taken $n$ times)

Calculation of higher-order derivatives

- For a polynomial function $f(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0$ $f (x) = a_{n} x^{n} + a_{n - 1} x^{n - 1} + \dots + a_{1} x + a_{0}$
  - Second derivative $f''(x) = n(n-1)a_nx^{n-2} + (n-1)(n-2)a_{n-1}x^{n-3} + \cdots + 2a_2$ (coefficients multiplied by decreasing powers of $x$ )
  - Third derivative $f'''(x) = n(n-1)(n-2)a_nx^{n-3} + (n-1)(n-2)(n-3)a_{n-1}x^{n-4} + \cdots + 6a_3$ (coefficients multiplied by further decreasing powers of $x$ )
- For an exponential function $f(x) = e^x$ $f (x) = e^{x}$
  - Second derivative $f''(x) = e^x$ (same as the original function)
  - Third derivative $f'''(x) = e^x$ (same as the original function)
  - $n$ th derivative $f^{(n)}(x) = e^x$ (same as the original function for all higher-order derivatives)
- For $f(x) = \sin(x)$ $f (x) = sin (x)$
  - Second derivative $f''(x) = -\sin(x)$ (negative of the original function)
  - Third derivative $f'''(x) = -\cos(x)$ (negative cosine of $x$ )
  - Fourth derivative $f^{(4)}(x) = \sin(x)$ (back to the original function)
- For $f(x) = \cos(x)$ $f (x) = cos (x)$
  - Second derivative $f''(x) = -\cos(x)$ (negative of the original function)
  - Third derivative $f'''(x) = \sin(x)$ (sine of $x$ )
  - Fourth derivative $f^{(4)}(x) = \cos(x)$ (back to the original function)

Patterns in derivative sequences

Polynomial functions
- $n$ th derivative of a polynomial function of degree $m$ equals zero if $n > m$ (derivatives eventually become zero)
- $n$ th derivative of a polynomial function of degree $m$ results in a polynomial function of degree $m-n$ if $n \leq m$ (degree decreases with each derivative)
Trigonometric functions
- Derivatives of sine and cosine functions follow a cyclic pattern
  1. $\sin(x)$
  2. $\cos(x)$
  3. $-\sin(x)$
  4. $-\cos(x)$
  5. $\sin(x)$ (pattern repeats)
- $n$ th derivative of $\sin(x)$ can be expressed as $\sin(x + \frac{n\pi}{2})$ (phase shift of $\frac{n\pi}{2}$ )
- $n$ th derivative of $\cos(x)$ can be expressed as $\cos(x + \frac{n\pi}{2})$ (phase shift of $\frac{n\pi}{2}$ )

Applications in physics and engineering

Acceleration
- Second derivative of position with respect to time
- If $s(t)$ represents the position of an object at time $t$ , then $s''(t)$ represents the acceleration at time $t$
Jerk
- Third derivative of position with respect to time or the rate of change of acceleration
- If $s(t)$ represents the position of an object at time $t$ , then $s'''(t)$ represents the jerk at time $t$
Other applications
- Analyzing the curvature of a function using higher-order derivatives
- Determining local maxima, local minima, and in optimization problems using higher-order derivatives

Key Terms to Review (21)

Acceleration: Acceleration is the rate of change of velocity of an object with respect to time. It reflects how quickly an object is speeding up, slowing down, or changing direction. This concept is crucial as it connects to the behavior of moving objects and helps in understanding their dynamics, which involves analyzing slopes, rates of change, and even the implications of higher-order derivatives.

Concavity: Concavity refers to the direction in which a function curves, either concave up or concave down. A function is concave up on an interval if its second derivative is positive, indicating that the slope of the tangent line is increasing, while it is concave down if its second derivative is negative, indicating that the slope is decreasing. Understanding concavity helps identify the behavior of a function, particularly in determining inflection points and analyzing the nature of extrema.

Curve sketching: Curve sketching is the process of analyzing the behavior and characteristics of a function to create a visual representation of its graph. This involves determining important features such as intercepts, increasing and decreasing intervals, local maxima and minima, concavity, and points of inflection. Understanding how higher-order derivatives influence these characteristics is essential for accurately sketching curves.

Differentiable functions: Differentiable functions are those that have a derivative at every point in their domain, which means they exhibit a defined rate of change and are locally linear. This property is crucial for understanding how functions behave, particularly in determining tangents, slopes, and higher-order derivatives. A function's differentiability also implies continuity, but not vice versa, making it a foundational concept for exploring more complex behaviors in calculus.

Exponential Functions: Exponential functions are mathematical functions of the form $$f(x) = a imes b^{x}$$, where $$a$$ is a constant, $$b$$ is a positive real number, and $$x$$ is the exponent. They describe processes that grow or decay at a constant rate proportional to their current value, making them crucial in modeling real-world phenomena such as population growth and radioactive decay.

F'''(x): The notation f'''(x) represents the third derivative of a function f with respect to the variable x. This derivative provides insight into the rate of change of the rate of change of the function, effectively measuring how the function's curvature changes over time. Understanding the third derivative is crucial for analyzing the behavior of functions, especially in determining points of inflection and concavity, as well as in applications like physics and engineering.

F''(x): The notation f''(x) represents the second derivative of a function f with respect to the variable x. It measures how the rate of change of a function's slope is changing, providing insight into the function's behavior, such as concavity and points of inflection, which are crucial for understanding the shape of graphs and optimizing functions.

F^{(n)}(x): The notation f^{(n)}(x) represents the n-th derivative of a function f with respect to the variable x. It signifies that the derivative has been taken n times, capturing how the behavior of the function changes as you differentiate it repeatedly. Higher-order derivatives are crucial in understanding the nature of functions, including their concavity, points of inflection, and overall shape.

Global extrema: Global extrema refer to the absolute maximum and minimum values of a function over its entire domain. These points are significant because they provide insights into the behavior of the function, indicating where it reaches its highest and lowest points, which is crucial in understanding the overall characteristics of the graph.

Inflection Points: Inflection points are points on a curve where the concavity changes, meaning the curve transitions from being concave up to concave down or vice versa. These points are important because they can indicate changes in the behavior of a function, such as transitions between increasing and decreasing rates of growth. Identifying inflection points involves using second derivatives to determine where they occur, providing insights into the function's overall shape and behavior.

Jerk: Jerk is the term used to describe the rate of change of acceleration, essentially measuring how quickly an object's acceleration is changing over time. It is the third derivative of position with respect to time, and it provides important insights into motion dynamics, especially in physics and engineering applications. Understanding jerk helps to analyze smoothness or roughness in motion, which is crucial in designing systems involving moving parts.

Leibniz Notation: Leibniz notation is a way of expressing derivatives using the symbols 'd' and the function's variable. It was developed by mathematician Gottfried Wilhelm Leibniz and is particularly useful for denoting both the first derivative and higher-order derivatives in a clear and systematic way. This notation helps to indicate the relationship between a function and its rates of change, making it essential for understanding concepts like higher-order derivatives.

Local extrema: Local extrema refer to the points on a function where the function reaches a local maximum or minimum value within a certain neighborhood. These points are crucial for understanding the behavior of functions, as they help identify where a function increases or decreases, and they often serve as indicators for optimization problems.

Mean Value Theorem: The Mean Value Theorem states that for a function that is continuous on a closed interval and differentiable on the open interval, there exists at least one point where the derivative equals the average rate of change of the function over that interval. This theorem provides a bridge between the behavior of a function and its derivatives, showing how slopes relate to overall changes.

Optimization: Optimization is the mathematical process of finding the maximum or minimum value of a function within a given set of constraints. This concept is crucial for identifying the best possible outcomes in various scenarios, such as maximizing profits or minimizing costs. Understanding optimization often involves analyzing derivatives to determine critical points, evaluating higher-order derivatives to assess the nature of these points, and applying specific methods to find solutions efficiently.

Polynomial Functions: Polynomial functions are mathematical expressions that represent relationships involving variables raised to whole number powers, where the coefficients can be real or complex numbers. They are continuous and smooth across their domain, making them crucial in calculus for understanding derivatives, integrals, and behavior of functions.

Second Derivative: The second derivative is the derivative of the derivative of a function, providing insight into the function's rate of change in relation to its own rate of change. This concept helps us understand not just how a function is changing, but also how the rate of that change is itself changing, revealing key features like concavity and potential inflection points. Additionally, the second derivative plays a significant role in analyzing the behavior of composite functions and the implications of implicit differentiation.

Smoothness: Smoothness refers to the property of a function that is continuous and has derivatives of all orders at every point in its domain. A function is considered smooth if it can be differentiated multiple times, and each of these derivatives is also continuous. This characteristic is crucial in understanding how functions behave and change, particularly when dealing with higher-order derivatives.

Taylor's Theorem: Taylor's Theorem is a fundamental result in calculus that provides an approximation of a function using polynomials based on the function's derivatives at a single point. This theorem essentially states that a sufficiently smooth function can be expressed as an infinite sum of terms calculated from the values of its derivatives at a specific point, allowing for the estimation of the function's value near that point. This concept is closely tied to higher-order derivatives, as the accuracy of the polynomial approximation depends on the number of derivatives used.

Third derivative: The third derivative is the derivative of the second derivative of a function, which provides insights into the rate of change of the rate of change of that function. This concept is part of higher-order derivatives and helps analyze the behavior and curvature of functions more thoroughly, revealing important characteristics such as inflection points and the overall shape of the graph.

Trigonometric Functions: Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides, commonly used in various fields like physics, engineering, and computer science. These functions include sine, cosine, tangent, cosecant, secant, and cotangent, and they play a vital role in understanding periodic phenomena. Their properties are essential when discussing derivatives, continuity, and the application of rules in calculus.

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Practice QuizGlossary

Practice Quiz Glossary

6.3 Higher-order derivatives

Higher-Order Derivatives

Concept of higher-order derivatives

Top images from around the web for Concept of higher-order derivatives

Top images from around the web for Concept of higher-order derivatives

Calculation of higher-order derivatives

Patterns in derivative sequences

Applications in physics and engineering

Key Terms to Review (21)

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AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.

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