Question

question find the derivative of $y = \frac{e^{3x^{2}}}{5x}$. select the correct answer below: $\frac{e^{3x^{2}}(x^{2}-1)}{25x^{2}}$, $\frac{5e^{3x^{2}}(6x^{2}-1)}{25x^{2}}$, $\frac{6xe^{3x^{2}} - 5}{25x^{2}}$, $\frac{e^{3x^{2}}-30x^{2}}{25x^{2}}$

Explanation:

Step1: Apply quotient - rule

$y=\frac{u}{v}$, $y^\prime=\frac{u^\prime v - uv^\prime}{v^{2}}$, where $u = e^{3x^{2}}$, $v = 5x$.

Step2: Find $u^\prime$ and $v^\prime$

$u^\prime=e^{3x^{2}}\cdot6x$, $v^\prime = 5$.

Step3: Substitute into quotient - rule

$y^\prime=\frac{e^{3x^{2}}\cdot6x\cdot5x - e^{3x^{2}}\cdot5}{(5x)^{2}}=\frac{30x^{2}e^{3x^{2}}- 5e^{3x^{2}}}{25x^{2}}$.

Answer:

$\frac{30x^{2}e^{3x^{2}}- 5e^{3x^{2}}}{25x^{2}}$