Question

What is the equation of the tangent line of `f(x)=(x-1)^2/(x+2)` at `x=3`?

Answer 1

`y=16/25x-28/25`

Explanation:

`•color(white)(x)m_(color(red)"tangent")=f'(x)" at x = a"`

`"differentiate using the "color(blue)"quotient rule"`

`"given "f(x)=(g(x))/(h(x))" then "`

`f'(x)=(h(x)g'(x)-g(x)h'(x))/(h(x))^2larr" quotient rule"`

`g(x)=(x-1)^2rArrg'(x)=2(x-1)`

`h(x)=x+2rArrh'(x)=1`

`rArrf'(x)=((x+2)2(x-1)-(x-1)^2)/(x+2)^2`

`rArrf'(3)=(20-4)/25=16/25`

`"and "f(3)=4/5`

`"using "m=16/25" and " (x_1,y_1)=(3,4/5)`

`rArry-4/5=16/25(x-3)`

`rArry=16/25x-28/25larr" equation of tangent"`