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If u(t) = < sin(5t), cos(7t), t > and v(t) = < t, cos(7t), sin(5t) >, use the formula below to find the given derivative. d/(dt)[u(t) cross v(t)] = u'\(t\) cross v(t) + u(t) cross v'\(t\)

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Final answer:

To calculate the derivative d/(dt)[u(t) cross v(t)], we differentiate the components of u(t) and v(t) with respect to time, then perform the cross product of these derivatives with the original vectors, and sum the results.

Step-by-step explanation:

In order to find the derivative d/(dt)[u(t) cross v(t)], we need to follow the given formula. This can be divided into two parts: finding u'(t) cross v(t), and finding u(t) cross v'(t). The vectors u(t) and v(t) have time-dependent components involving trigonometric functions. Hence, when we differentiate each component of u(t) and v(t) with respect to time t, we will apply basic calculus differentiation rules for sine and cosine functions.

We then proceed with the cross product of the differentiated vectors with the original vectors respectively. This would involve using the determinant method for calculating the cross product of two vectors in three-dimensional space. Summing up the results will give us the derivative of u(t) cross v(t) with respect to time.

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