Find the following higher-order derivatives.dn/dxn (2x)

Ch. 3 - Derivatives

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 3 - Derivatives

Problem 3.9.90

Chapter 3, Problem 3.9.90

Find the following higher-order derivatives.
dⁿ/dxⁿ (2^x)

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Identify the function for which you need to find the higher-order derivative. In this case, the function is f(x) = 2x.

Recognize that the first derivative of f(x) = 2x with respect to x is a constant. Use the power rule for differentiation: d/dx (2x) = 2.

Understand that the second derivative of a constant is zero. Therefore, d²/dx² (2) = 0.

Realize that for any n ≥ 2, the nth derivative of a constant is zero. Thus, for n ≥ 2, dⁿ/dxⁿ (2x) = 0.

Conclude that the higher-order derivatives of the function 2x, for n ≥ 2, are zero, as the function becomes a constant after the first derivative.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Higher-Order Derivatives

Higher-order derivatives refer to the derivatives of a function taken multiple times. The first derivative gives the rate of change of the function, the second derivative provides information about the curvature, and so on. For a function f(x), the n-th derivative is denoted as f^(n)(x) and is crucial for analyzing the behavior of functions in calculus.

Exponential Functions

Exponential functions are mathematical functions of the form f(x) = a^x, where 'a' is a constant and 'x' is the variable. The function 2^x is an example, where the base is 2. These functions are characterized by their rapid growth and unique properties, such as the fact that their derivative is proportional to the function itself, which simplifies the process of finding higher-order derivatives.

Leibniz Notation

Leibniz notation is a way of expressing derivatives using the symbols 'd' and 'dx'. For example, d^n/dx^n indicates the n-th derivative with respect to x. This notation is particularly useful in calculus for clearly denoting the order of differentiation and is essential for understanding the process of taking higher-order derivatives, especially in complex functions.

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