Question

How do you differentiate `f(x) =1/(e^(3x)-2)` using the quotient rule?

Answer 1

`f'(x) = frac{- 3*e^{3x}}{(e^{3x}-2)^2}`

Explanation:

The Quotient Rule

`frac{"d"}{"d"x}(u/v) = frac{v*frac{"d"u}{"d"x} - u*frac{"d"v}{"d"x}}{v^2}`

In this case, `u=1` and `v=e^{3x}-2`.

`f'(x) = frac{"d"}{"d"x}(1/(e^{3x}-2))`

`= frac{(e^{3x}-2)*frac{"d"}{"d"x}(1) - (1)*frac{"d"}{"d"x}(e^{3x}-2)}{(e^{3x}-2)^2}`

`= frac{(e^{3x}-2)*(0) - 3*e^{3x}}{(e^{3x}-2)^2}`

`= frac{- 3*e^{3x}}{(e^{3x}-2)^2}`