Question

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

Answer 1

`y=-5x+6`

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))^2`

`g(x)=-x^2-x+3rArrg'(x)=-2x-1`

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

`rArrf'(x)=((2x-1)(-2x-1)-2(-x^2-x+3))/(2x-1)^2`

`rArrf'(1)=(-3-2)/1=-5`

`rArrf(1)=1to(1,1)larr" tangent point"`

`rArry-1=-5(x-1)`

`rArry=-5x+6larr" equation of tangent"`