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If $y=\cos t$ and $x=\sin t$, then what is $\frac{d y}{d x}$ equal to?
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Verified Answer
The correct answer is:
$-x / y$
Let $y=\cos t, x=\sin t$
$\frac{\mathrm{dy}}{\mathrm{dt}}=-\sin \mathrm{t}, \frac{\mathrm{dx}}{\mathrm{dt}}=\cos \mathrm{t}$
$\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{dy} / \mathrm{dt}}{\mathrm{dx} / \mathrm{dt}}=-\frac{\sin \mathrm{t}}{\cos \mathrm{t}}=-\frac{\mathrm{x}}{\mathrm{y}}$
$\frac{\mathrm{dy}}{\mathrm{dt}}=-\sin \mathrm{t}, \frac{\mathrm{dx}}{\mathrm{dt}}=\cos \mathrm{t}$
$\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{dy} / \mathrm{dt}}{\mathrm{dx} / \mathrm{dt}}=-\frac{\sin \mathrm{t}}{\cos \mathrm{t}}=-\frac{\mathrm{x}}{\mathrm{y}}$
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