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# r(t) = ti + 3tj + t²k, u(t) = 4ti + t²j + t³k Use the properties of the derivative to find the following. (a)r′(t) (b)d/dt [3r(t) - u(t)] (c)d/dt (5t)u(t) (d)d/dt [r(t)·u(t)] (e)d/dt [r(t)×u(t)] (f)d/dt r(2t)

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In this exercise, we need to use properties of the derivative to find the values of the derivative of combinations of the given vector-valued functions:

$\mathbf{r}(t)=f_1(t)\mathbf{i}+g_1(t)\mathbf{j}+h_1(t)\mathbf{k},$

where $f_1(t)=t,\,g_1(t)=3t,\,h_1(t)=t^2$, and

$\mathbf{u}(t)=f_2(t)\mathbf{i}+g_2(t)\mathbf{j}+h_2(t)\mathbf{k},$

where $f_2(t)=4t,\,g_2(t)=t^2,\,h_2(t)=t^3.$

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