Discussion
If \(y(x) = 15\cos(x) - 13\sin(x)\), then \(\frac{d^{2}y}{dx^{2}}\) will be
Answer: C
The second derivative is \(-y\).
Differentiate twice, using \(\frac{d}{dx}\cos x = -\sin x\) and \(\frac{d}{dx}\sin x = \cos x\):
\[ \begin{aligned} y &= 15\cos x - 13\sin x \\ \frac{dy}{dx} &= -15\sin x - 13\cos x \\ \frac{d^{2}y}{dx^{2}} &= -15\cos x + 13\sin x \\ &= -(15\cos x - 13\sin x) = -y \end{aligned} \]
Any combination of \(\sin x\) and \(\cos x\) satisfies \(y'' = -y\), which is the equation of simple harmonic motion. The constants 15 and 13 do not matter. The result is not a constant such as 2, and it contains no \(\pi\) or \(x\) in a denominator, so the other three options are ruled out.
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