# How do you use the definition of a derivative to find the derivative of f(x) = 7/(9x)?

Mar 7, 2016

For all ${x}_{0} \in \mathbb{R} \text{\} \left\{0\right\} : f ' \left({x}_{0}\right) = - \frac{7}{9 {x}_{0}^{2}}$.

#### Explanation:

For all ${x}_{0} \in \mathbb{R} \text{\} \left\{0\right\} :$

$f ' \left({x}_{0}\right) = {\lim}_{x \to {x}_{0}} \frac{f \left(x\right) - f \left({x}_{0}\right)}{x - {x}_{0}}$

$= {\lim}_{x \to {x}_{0}} \frac{\left(\frac{7}{9 x}\right) - \left(\frac{7}{9 {x}_{0}}\right)}{x - {x}_{0}}$

$= {\lim}_{x \to {x}_{0}} \frac{\frac{7 {x}_{0} - 7 x}{9 x \cdot {x}_{0}}}{x - {x}_{0}}$

$= {\lim}_{x \to {x}_{0}} \frac{- 7 \left(x - {x}_{0}\right)}{\left(9 x \cdot {x}_{0}\right) \cdot \left(x - {x}_{0}\right)}$

$= {\lim}_{x \to {x}_{0}} - \frac{7}{9 x \cdot {x}_{0}}$

$= - \frac{7}{9 {x}_{0}^{2}} .$