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.

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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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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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