differentiate. f(x)=e^4x/x^16 f(x)=

Explanation:

Step1: Recall quotient - rule

The quotient - rule states that if $F(x)=\frac{u(x)}{v(x)}$, then $F^{\prime}(x)=\frac{u^{\prime}(x)v(x)-u(x)v^{\prime}(x)}{v^{2}(x)}$. Here, $u(x) = e^{4x}$ and $v(x)=x^{16}$.

Step2: Find $u^{\prime}(x)$

Using the chain - rule, if $y = e^{4x}$, let $u = 4x$, then $\frac{dy}{du}=e^{u}$ and $\frac{du}{dx}=4$. So, $u^{\prime}(x)=\frac{d}{dx}(e^{4x})=4e^{4x}$.

Step3: Find $v^{\prime}(x)$

Using the power - rule $\frac{d}{dx}(x^{n})=nx^{n - 1}$, for $v(x)=x^{16}$, $v^{\prime}(x)=16x^{15}$.

Step4: Apply quotient - rule

$F^{\prime}(x)=\frac{4e^{4x}\cdot x^{16}-e^{4x}\cdot16x^{15}}{(x^{16})^{2}}$.

Step5: Simplify the expression

Factor out $4e^{4x}x^{15}$ from the numerator: $F^{\prime}(x)=\frac{4e^{4x}x^{15}(x - 4)}{x^{32}}=\frac{4e^{4x}(x - 4)}{x^{17}}$.

Answer:

$\frac{4e^{4x}(x - 4)}{x^{17}}$