How do you use the definition of a derivative to find the derivative of $f (x) = \frac{7}{9 x}$ $f(x) = 7/(9x)$ ?

Question

1871 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-03-07T16:21:44

For all $x_0 in RR "\" {0} : f'(x_0) = -7/(9x_0^2)$ .
See answer below.

For all $x_0 in RR "\" {0} :$

$f'(x_0) = lim_(x->x_0)(f(x)-f(x_0))/(x-x_0)$

$= lim_(x->x_0)((7/(9x))-(7/(9x_0)))/(x-x_0)$

$= lim_(x->x_0)((7x_0-7x)/(9x*x_0))/(x-x_0)$

$= lim_(x->x_0)(-7(x-x_0))/((9x*x_0)*(x-x_0))$

$= lim_(x->x_0)-7/(9x*x_0)$

$=-7/(9x_0^2).$

How do you use the definition of a derivative to find the derivative of f(x) = 7/(9x)f(x)=79xf(x) = 7/(9x)?