-x to the -1 power
$$(-x)^{-1} \\\\
=\dfrac{1}{(-x)}\\\\
=\dfrac{1}{-x}\\ \\
=-\dfrac{1}{x}$$
Use the divison rule for exponents.
$${\frac{{{\mathtt{x}}}^{{\mathtt{m}}}}{{{\mathtt{x}}}^{{\mathtt{n}}}}} = {{\mathtt{x}}}^{\left({\mathtt{m}}{\mathtt{\,-\,}}{\mathtt{n}}\right)}$$
Just like when you write a fraction in simplest form ie $${\frac{{\mathtt{3}}}{{\mathtt{2}}}}$$, we rewrite this as 1 1/2.
In the final answer we can not report negitive exponents.
We take the smallest exponent across the divisor bar and subtract from the quanity currently there.
For example: $${{\mathtt{\,-\,}}{\mathtt{x}}}^{-{\mathtt{1}}}$$ can be rewritten as $${\frac{\left({{\mathtt{\,-\,}}{\mathtt{x}}}^{-{\mathtt{1}}}\right)}{{{\mathtt{x}}}^{{\mathtt{0}}}}}$$ then we move -1 across the divisor bar because it is less than 0, then we have $${\mathtt{\,-\,}}{\frac{\left({\mathtt{1}}\right)}{\left({{\mathtt{x}}}^{\left({\mathtt{0}}{\mathtt{\,\small\textbf+\,}}{\mathtt{1}}\right)}\right)}}$$ you are now asking why is it $${{\mathtt{x}}}^{\left({\mathtt{0}}{\mathtt{\,\small\textbf+\,}}{\mathtt{1}}\right)}$$, it is because when we subtract a negitive number we actually add it. Which makes it,$${\mathtt{\,-\,}}{\frac{{\mathtt{1}}}{{{\mathtt{x}}}^{{\mathtt{1}}}}}$$, since we don't normally express 1 as an exponent.
Which makes our final answer:
$${\mathtt{\,-\,}}{\frac{{\mathtt{1}}}{{\mathtt{x}}}}$$