# How do you find the derivative of 1/(2x)?

Jul 9, 2016

$- \frac{1}{2 {x}^{2}}$

#### Explanation:

Differentiate using the $\textcolor{b l u e}{\text{power rule}}$

$\textcolor{red}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{\frac{d}{\mathrm{dx}} \left(a {x}^{n}\right) = n a {x}^{n - 1}} \textcolor{w h i t e}{\frac{a}{a}} |}}}$

Rewrite the function as.

$\frac{1}{2 x} = \frac{1}{2} \times \frac{1}{x} = \frac{1}{2} \times {x}^{-} 1 = \frac{1}{2} {x}^{-} 1$

$\Rightarrow \frac{d}{\mathrm{dx}} \left(\frac{1}{2} {x}^{-} 1\right) = - 1 \times \frac{1}{2} {x}^{- 1 - 1} = - \frac{1}{2} {x}^{-} 2$

$\Rightarrow \frac{d}{\mathrm{dx}} \left(\frac{1}{2 x}\right) = - \frac{1}{2} {x}^{-} 2 = - \frac{1}{2 {x}^{2}}$

Jul 9, 2016

ALTERNATIVE APPROACH

#### Explanation:

By the quotient rule:

$\left(\frac{1}{2 x}\right) ' = \frac{\left(0 \times 2 x\right) - \left(1 \times 2\right)}{2 x} ^ 2$

$\left(\frac{1}{2 x}\right) ' = - \frac{2}{4 {x}^{2}}$

$\left(\frac{1}{2 x}\right) ' = - \frac{1}{2 {x}^{2}}$

Hopefully this helps!