differentiate, f(x) = e^3x / x^15, f(x) =

Explanation:

Step1: Recall quotient - rule

The quotient - rule states that if $F(x)=\frac{u(x)}{v(x)}$, then $F^{\prime}(x)=\frac{u^{\prime}(x)v(x)-u(x)v^{\prime}(x)}{v(x)^2}$. Here, $u(x) = e^{3x}$ and $v(x)=x^{15}$.

Step2: Differentiate $u(x)$

Using the chain - rule, if $y = e^{3x}$, let $t = 3x$, then $\frac{dy}{dt}=e^{t}$ and $\frac{dt}{dx}=3$. So, $u^{\prime}(x)=\frac{d}{dx}(e^{3x})=3e^{3x}$.

Step3: Differentiate $v(x)$

Using the power - rule $\frac{d}{dx}(x^n)=nx^{n - 1}$, for $v(x)=x^{15}$, we have $v^{\prime}(x)=15x^{14}$.

Step4: Apply quotient - rule

$F^{\prime}(x)=\frac{3e^{3x}\cdot x^{15}-e^{3x}\cdot15x^{14}}{(x^{15})^2}$.

Step5: Simplify the expression

Factor out $3e^{3x}x^{14}$ from the numerator: $F^{\prime}(x)=\frac{3e^{3x}x^{14}(x - 5)}{x^{30}}=\frac{3e^{3x}(x - 5)}{x^{16}}$.

Answer:

$\frac{3e^{3x}(x - 5)}{x^{16}}$