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Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 3 - DerivativesProblem 3.5.9
Chapter 3, Problem 3.5.9

Find d²/dx² (sin x + cos x).

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1
First, identify the function you need to differentiate: f(x) = sin(x) + cos(x).
Calculate the first derivative, f'(x), by differentiating each term separately. The derivative of sin(x) is cos(x), and the derivative of cos(x) is -sin(x). Therefore, f'(x) = cos(x) - sin(x).
Now, find the second derivative, f''(x), by differentiating f'(x). Differentiate cos(x) to get -sin(x) and differentiate -sin(x) to get -cos(x).
Combine the results from the previous step to express the second derivative: f''(x) = -sin(x) - cos(x).
Review the process to ensure each differentiation step was applied correctly, confirming that the second derivative of the original function is f''(x) = -sin(x) - cos(x).

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Key Concepts

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

Differentiation

Differentiation is the process of finding the derivative of a function, which represents the rate of change of the function with respect to a variable. In this context, we need to differentiate the function sin x + cos x to find its first derivative, which will then be differentiated again to obtain the second derivative.
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Second Derivative

The second derivative of a function is the derivative of the first derivative. It provides information about the curvature of the function and can indicate concavity. In this case, calculating d²/dx² (sin x + cos x) involves taking the derivative of the first derivative to analyze how the rate of change itself is changing.
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Trigonometric Derivatives

Trigonometric derivatives are specific rules for differentiating trigonometric functions. For example, the derivative of sin x is cos x, and the derivative of cos x is -sin x. Understanding these derivatives is essential for solving the given problem, as they will be applied to find both the first and second derivatives of the function sin x + cos x.
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Derivatives of Other Inverse Trigonometric Functions
Related Practice
Textbook Question

Derivatives Find and simplify the derivative of the following functions.

h(x) = (x−1)(2x²-1) / (x³-1)

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Textbook Question

27–76. Calculate the derivative of the following functions.

x2+93\(\sqrt\)[3]{x^2+9}

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Textbook Question

Calculate the derivative of the following functions. In some cases, it is useful to use the properties of logarithms to simplify the functions before computing f'(x).

f(x) = In(sec⁴x tan² x)

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Textbook Question

Evaluate the following limits or state that they do not exist. (Hint: Identify each limit as the derivative of a function at a point.)

lim x→π/2 cos x/x−π/2

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Textbook Question

Continuity of a piecewise function Let g(x) = <matrix 2x1> For what values of a is g continuous?

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Textbook Question

5–8. Calculate dy/dx using implicit differentiation.

x = y²

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