If $x<0$, then which of the following must be positive?
$\textbf{(A)}\ \frac{x}{\left|x\right|} \qquad \textbf{(B)}\ -x^2 \qquad \textbf{(C)}\ -2^x \qquad \textbf{(D)}\ -x^{-1} \qquad \textbf{(E)}\ \sqrt[3]{x}$
$x$ is negative, so we can just place a negative value into each expression and find the one that is positive. Suppose we use $-1$.
$\textbf{(A)} \Rightarrow \frac{-1}{|-1|} = -1$
$\textbf{(B)} \Rightarrow -(-1)^2 = -1$
$\textbf{(C)} \Rightarrow -2^{(-1)} = -\frac{1}{2}$
$\textbf{(D)} \Rightarrow -(-1)^{(-1)} = 1$
$\textbf{(E)} \Rightarrow \sqrt[3]{-1} = -1$

Obviously only $\boxed{\textbf{(D)}}$ is positive.