Question

How do you differentiate `f(x)= 1/(e^(3x) -5x)` using the quotient rule?

Answer 1

`(5-3e^(3x))/( e^(3x) - 5x)^2`

Explanation:

The quotient Rule states that
the derivative of a division of two functions `f(x)/g(x)`
Is equal to `(f'(x)g(x) - f(x)g'(x))/ (g(x))^2`

therefore in your Question,
let `f(x) = 1`
and `g(x) = e^(3x) - 5x`

their respective derivatives are
`f'(x) = 0`
`g'(x) = 3e^(3x) - 5`

therefore, the derivative of the entire equation using the Quotient rule is
`((0*e^(3x) - 5x) - (1*(3e^(3x) - 5)))/ ( e^(3x) - 5x)^2`

`= - (3e^(3x) - 5)/( e^(3x) - 5x)^2`

which is equal to
`(5-3e^(3x))/( e^(3x) - 5x)^2`
=
and you can simplify it more if you want but that's basically it