Question

How do you differentiate `f(x) = (5x)/(3x^2-4x+6)` using the quotient rule?

Answer 1

`f'(x)=(-15x^2+30)/(3x^2-4x+6)^2`

Explanation:

For a function `f(x)=(g(x))/(h(x))`, the quotient rule states that the derivative of the function is

`f'(x)=(g'(x)h(x)-g(x)h'(x))/[h(x)]^2`

In this scenario, we know the following:

`g(x)=5xcolor(white)(xxxx)=>color(white)(xxxx)g'(x)=5`

`h(x)=3x^2-4x+6color(white)(xxxx)=>color(white)(xxxx)h'(x)=6x-4`

Thus, plugging these into the quotient rule formula,

`f'(x)=(5(3x^2-4x+6)-5x(6x-4))/(3x^2-4x+6)^2`

Simplify.

`f'(x)=(15x^2-20x+30-30x^2+20x)/(3x^2-4x+6)^2`

`f'(x)=(-15x^2+30)/(3x^2-4x+6)^2`