The derivative of $\dfrac1{x^2}$ is $\boxed{-\dfrac2{x^3}}$.

To find the derivative of $\dfrac1{x^2}$, we will use the Power Rule:

\[\begin{align*} \dfrac{d}{dx} x^n &= nx^{n – 1} \end{align*}\]

Notice that $\dfrac1{x^2}$ may be rewritten as $x^{-2}$, making the use of the power rule quite straightforward.

\[\begin{align*} \dfrac{d}{dx} \dfrac1{x^2} &= \dfrac{d}{dx} x^{-2}\\ &= -2x^{-2 – 1}\\ &= -2x^{-3}\\ &= -\dfrac2{x^3} \end{align*}\]

Thus, by the power rule, the derivative of $\dfrac1{x^2}$ is $\boxed{-\dfrac2{x^3}}$.

Note that the power rule is very handy when dealing with derivatives that involve exponents so it is a good formula to keep in the back of your mind. It allows you to quickly find the derivative of any term with $x$ raised to either a positive or negative power.

### Similar Example

Find the derivative of $\dfrac{1}{x^{3/2}}$.

First, rewrite the Function as $x^{-3/2}$:

\[\begin{align*} \dfrac{1}{x^{3/2}} = x^{-3/2} \end{align*}\]

Next, apply the Power Rule to $x^{-3/2}$.

\[\begin{align*} \frac{d}{dx} x^{-3/2} = -\dfrac{3}{2} \cdot x^{-5/2} \end{align*}\]

Finally, if needed convert the negative exponent back to a positive one.

\[\begin{align*} -\dfrac{3}{2} \cdot x^{-5/2} = -\dfrac{3}{2x^{5/2}} \end{align*}\]

Final Answer:

\[\begin{align*} \boxed{\displaystyle \frac{d}{dx} \left( \dfrac{1}{x^{3/2}} \right) = -\dfrac{3}{2x^{5/2}}} \end{align*}\]