Question

Compute the derivative for r(t) = (cos 5t)i + (sin⁸ t)] + (tan 9t)k.
dr/dt=

Answer 1

To find the derivative dr/dt for r(t) = (cos 5t)i + (sin⁸ t) + (tan 9t)k, we differentiate each component separately, resulting in -5 sin(5t)i + 8(sin⁷ t)(cos t) + 9 sec²(9t)k.

Step-by-step explanation:

To compute the derivative dr/dt for the function r(t) = (cos 5t)i + (sin⁸ t) + (tan 9t)k, we need to take the derivative of each component of this vector-valued function with respect to time t. We apply the rules of differentiation to each component separately.

For the first component (cos 5t)i, the derivative is -5 sin(5t)i, using the chain rule where the derivative of cos u with respect to u is -sin u and d(5t)/dt = 5.

For the second component (sin⁸ t), using the power rule and chain rule, we get 8(sin⁷ t)(cos t).

For the third component (tan 9t)k, using the derivative of tan u with respect to u is sec² u and d(9t)/dt = 9, we obtain 9 sec²(9t)k.

Combining these, dr/dt is -5 sin(5t)i + 8(sin⁷ t)(cos t) + 9 sec²(9t)k.