Question

determine the derivative of $f(x)=\frac{e^{3x}}{x^{6}}$. select the correct answer below: $\frac{3e^{3x}(x^{2}-2)}{x^{7}}$, $\frac{e^{3x}(x^{2}-6)}{x^{7}}$, $\frac{3e^{3x}(x - 2)}{x^{7}}$, $\frac{e^{3x}(x - 6)}{x^{7}}$

Explanation:

Step1: Apply quotient - rule

The quotient - rule states that if $y=\frac{u}{v}$, then $y^\prime=\frac{u^\prime v - uv^\prime}{v^{2}}$. Here, $u = e^{3x}$, $u^\prime=3e^{3x}$, $v = x^{6}$, and $v^\prime = 6x^{5}$.

Step2: Substitute values

$f^\prime(x)=\frac{3e^{3x}\cdot x^{6}-e^{3x}\cdot6x^{5}}{(x^{6})^{2}}$.

Step3: Simplify

$f^\prime(x)=\frac{e^{3x}(3x^{6}-6x^{5})}{x^{12}}=\frac{3e^{3x}(x - 2)}{x^{7}}$.

Answer:

$\frac{3e^{3x}(x - 2)}{x^{7}}$ (corresponding to the third option)