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Compute derivatives of F(t) = log (1+t2) i + arctan t j + 1/(1+t2) k

Compute the derivatives F'(t) and F''(t) of the vector valued function

    \[ F(t) = \log(1+t^2) \mathbf{i} + \arctan t \mathbf{j} + \frac{1}{1+t^2} \mathbf{k}. \]


We compute

    \begin{align*}  F'(t) &= \left( \frac{2t}{1+t^2}, \frac{1}{1+t^2}, \frac{-2t}{(1+t^2)^2} \right) \\[9pt]  F''(t) &= \left( \frac{2(1-t^2)}{(1+t^2)^2}, \frac{-2t}{(1+t^2)^2}, \frac{6t^2 - 2}{(1+t^2)^2} \right). \end{align*}

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