# The position of a particle moving along the x-axis for any time, t> 0, is given by s(t) = - cos^2 (2t) + ln(t) + e ^ sin(t) where s(t) is measured in inches and t is measured in seconds.

Find the velocity of the particle at t = 8 seconds.

## s(t) = - cos^2 (2t) + ln(t) + e ^ sin(t)

now just use the chain rule to find
v(t) = -2cos(2t)*(-sin(2t))*2 + 1/t + e^sint * cost
now find v(8)

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## To find the velocity of the particle at t = 8 seconds, we need to take the derivative of the position function, s(t), with respect to time, t. Let's proceed with the differentiation:

s(t) = -cos^2(2t) + ln(t) + e^sin(t)

To find the velocity, we take the derivative of s(t) with respect to t:

v(t) = d/dt[-cos^2(2t) + ln(t) + e^sin(t)]

First, we'll differentiate -cos^2(2t):

Using the chain rule, the derivative of -cos^2(2t) is 2cos(2t) * -sin(2t) * 2 = -4cos(2t)sin(2t)

Next, we'll differentiate ln(t):

The derivative of ln(t) is 1/t.

Finally, we'll differentiate e^sin(t):

Using the chain rule, the derivative of e^sin(t) is cos(t) * e^sin(t).

Now we can write the derivative of s(t), v(t), as:

v(t) = -4cos(2t)sin(2t) + 1/t + cos(t) * e^sin(t)

To find the velocity of the particle at t = 8 seconds, we substitute t = 8 into the velocity function:

v(8) = -4cos(2(8))sin(2(8)) + 1/8 + cos(8) * e^sin(8)

Now we can evaluate this expression using a calculator to find the velocity at t = 8 seconds.

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## To find the velocity of the particle at t = 8 seconds, we need to differentiate the position function, s(t), with respect to time, t.

The position function is given as:

s(t) = -cos^2 (2t) + ln(t) + e^sin(t)

To differentiate s(t), we need to apply the rules of differentiation to each term individually.

1. Differentiating -cos^2 (2t):

To differentiate this term, we need to apply the chain rule and the power rule of differentiation.

Derivative of -cos^2 (2t):
= -2cos(2t) * -sin(2t) * 2
= 4cos(2t)sin(2t)

2. Differentiating ln(t):

To differentiate this term, we use the derivative of the natural logarithm which is 1/t.

Derivative of ln(t):
= 1/t

3. Differentiating e^sin(t):

To differentiate this term, we use the chain rule and the derivative of the sine function.

Derivative of e^sin(t):
= e^sin(t) * cos(t)

Now, we can find the velocity at t = 8 seconds by substituting t = 8 into the derivative of s(t):

Velocity at t = 8 seconds:
= 4cos(2*8)sin(2*8) + 1/8 + e^sin(8) * cos(8)
= 4cos(16)sin(16) + 1/8 + e^sin(8) * cos(8)

Calculating the exact numerical value for the velocity at t = 8 seconds would require plugging in the specific values for cos(16), sin(16), and e^sin(8).