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

Mar 3, 2018

$\frac{5 - 3 {e}^{3 x}}{{e}^{3 x} - 5 x} ^ 2$

#### Explanation:

The quotient Rule states that
the derivative of a division of two functions $f \frac{x}{g} \left(x\right)$
Is equal to $\frac{f ' \left(x\right) g \left(x\right) - f \left(x\right) g ' \left(x\right)}{g \left(x\right)} ^ 2$

let $f \left(x\right) = 1$
and $g \left(x\right) = {e}^{3 x} - 5 x$

their respective derivatives are
$f ' \left(x\right) = 0$
$g ' \left(x\right) = 3 {e}^{3 x} - 5$

therefore, the derivative of the entire equation using the Quotient rule is
$\frac{\left(0 \cdot {e}^{3 x} - 5 x\right) - \left(1 \cdot \left(3 {e}^{3 x} - 5\right)\right)}{{e}^{3 x} - 5 x} ^ 2$

$= - \frac{3 {e}^{3 x} - 5}{{e}^{3 x} - 5 x} ^ 2$

which is equal to
$\frac{5 - 3 {e}^{3 x}}{{e}^{3 x} - 5 x} ^ 2$
=
and you can simplify it more if you want but that's basically it