Question 1204825
Estimate the slope of the tangent line to the graph of {{{f(x)=x/(x-2)}}} at the point where {{{x = 3}}}

first find that point

 {{{f(3)=3/(3-2)=3/1=3}}}
 the point where {{{x = 3}}} is ({{{3}}},{{{3}}})

To find the slope of the tangent line at the point ({{{3}}},{{{3}}}), we need to take the derivative of {{{f(x) =x/(x-2)}}}  which is {{{y}}}'{{{(x) = -2/(x - 2)^2}}}


Therefore, the slope of the tangent line at the point ({{{3}}},{{{3}}}) is 
 
{{{y}}}'{{{(3) = -2/(3 - 2)^2=-2/1=-2}}}


equation of the tangent will be

{{{y=-2x+b}}}...use point ({{{3}}},{{{3}}}) to find {{{b}}}

{{{3=-2*3+b}}}

{{{b=3+6}}}

{{{b=9}}}


and equation of the tangent is

{{{y=-2x+9}}}


{{{ drawing( 600, 600, -10, 10, -10, 10,
circle(3,3,.12), locate(3,3,p(3,3)),
graph( 600, 600, -10, 10, -10, 10, x/(x-2), -2x+9)) }}}