Question

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

Explanation:

Step1: Apply quotient - rule

The quotient - rule states that if $y=\frac{u}{v}$, then $y'=\frac{u'v - uv'}{v^{2}}$. Here, $u = e^{3x^{2}}$ and $v = 5x$. First, find $u'$ using the chain - rule. If $u = e^{3x^{2}}$, let $t = 3x^{2}$, then $\frac{du}{dt}=e^{t}$ and $\frac{dt}{dx}=6x$, so $u'=6xe^{3x^{2}}$, and $v' = 5$.

Step2: Substitute into quotient - rule

$y'=\frac{6xe^{3x^{2}}\cdot5x - e^{3x^{2}}\cdot5}{(5x)^{2}}=\frac{30x^{2}e^{3x^{2}}-5e^{3x^{2}}}{25x^{2}}=\frac{5e^{3x^{2}}(6x^{2}-1)}{25x^{2}}$

Answer:

$\frac{5e^{3x^{2}}(6x^{2}-1)}{25x^{2}}$ (corresponding to the third option)