Textbook Question
Find f′(x), f′′(x), and f′′′(x) for the following functions.
f(x) = 3x2 + 5ex
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3. Techniques of Differentiation
Higher Order Derivatives
1:46 minutesProblem 3.5.65.b
Textbook Question
Explain why or why not Determine whether the following statements are true and give an explanation or counter example.
b. d²/dx² (sin x) = sin x.
Verified step by step guidance1
To determine if the statement \( \frac{d^2}{dx^2} (\sin x) = \sin x \) is true, we need to find the second derivative of \( \sin x \).
First, find the first derivative of \( \sin x \). The derivative of \( \sin x \) with respect to \( x \) is \( \cos x \).
Next, find the second derivative by differentiating \( \cos x \). The derivative of \( \cos x \) with respect to \( x \) is \( -\sin x \).
Thus, the second derivative of \( \sin x \) is \( -\sin x \), not \( \sin x \).
Therefore, the statement \( \frac{d^2}{dx^2} (\sin x) = \sin x \) is false. The correct second derivative is \( -\sin x \).
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1mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Second Derivative
The second derivative of a function measures the rate of change of the first derivative. It provides information about the concavity of the function and can indicate points of inflection. In this context, calculating the second derivative of sin x involves differentiating the function twice with respect to x.
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Trigonometric Functions
Trigonometric functions, such as sine and cosine, are periodic functions that relate angles to ratios of sides in right triangles. The sine function, sin x, oscillates between -1 and 1 and has a specific pattern of derivatives: the first derivative is cos x, and the second derivative is -sin x.
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True/False Statements in Calculus
In calculus, determining the truth of a statement often involves verifying mathematical identities or properties. For the statement d²/dx² (sin x) = sin x, one must compute the second derivative and compare it to sin x to establish its validity, which in this case reveals that the statement is false.
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