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## Q. 3.3

Let $\mathbf{f}(t)=(\ln t) \mathbf{i}+(1 / t) \mathbf{j} \text {. Calculate } \mathbf{f}^{\prime \prime}(t)$.

## Verified Solution

\begin{aligned} \mathbf{f}^{\prime}(t)=(1 / t) \mathbf{i}-\left(1 / t^{2}\right) \mathbf{j}, \text { so } \mathbf{f}^{\prime \prime}(t)=-\frac{1}{t^{2}} \mathbf{i}+\frac{2}{t^{3}} \mathbf{j}\text {.}\end{aligned}

Note that $\mathbf{f}^{\prime}$ and $\mathbf{f}^{\prime \prime}$  are defined for all t>0. ( $\mathbf{f}^{\prime}$ and $\mathbf{f}^{\prime \prime}$ are not defined for $t \leq 0$ because $\ln t$ is not defined for $t \leq 0$.)