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The displacement of a particle on a vibrating string is given by the equation $ s(t) = 10 + \frac {1}{4} \sin (10 \pi t) $ where $ s $ is measured in centimeters and $ t $ in seconds. Find the velocity of the particle after $ t $ seconds.

$\frac{5 x}{2} \cos (10 \pi t) \mathrm{cm} / \mathrm{s}$

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All right, since we are given S of T uh is 10 plus a 1/4 signed of 10 pi t. The theme in this problem is when they ask for velocity, that's the same thing as the derivative of that. Uh and just a reminder that the derivative of a constant is zero. I'll go and write that for right now. And then what we have going on here is the chain rule where the first thing that you do is you take the derivative of the outer function. So the derivative of sine would be the outer function is co sign. And what you do first is and leave the inner function alone. But I have to multiply by the derivative of the inner function. And because T is your independent variable, It's just a constant times t. So the derivative of that would be 10 pie. So when you go to clean up your answer, You know, 10/4 reduces to 5/2. You can stick that pie in the new numerator. Tell you multiply fractions Of 10 PT. Co sign of temp it and that's your answer for velocity and we're done at a unit to it are good.