Question

use the quotient rule to find the derivative of the function.

$g(x)=\frac{sin(x)}{x^{7}}$

Explanation:

Step1: Recall quotient - rule formula

The quotient - rule states that if $g(x)=\frac{f(x)}{h(x)}$, then $g^\prime(x)=\frac{f^\prime(x)h(x)-f(x)h^\prime(x)}{[h(x)]^2}$. Here, $f(x)=\sin(x)$ and $h(x)=x^{7}$.

Step2: Find $f^\prime(x)$ and $h^\prime(x)$

The derivative of $f(x)=\sin(x)$ is $f^\prime(x)=\cos(x)$, and the derivative of $h(x)=x^{7}$ using the power - rule $(x^n)^\prime=nx^{n - 1}$ is $h^\prime(x)=7x^{6}$.

Step3: Apply the quotient - rule

Substitute $f(x), f^\prime(x), h(x), h^\prime(x)$ into the quotient - rule formula:
\[

$$\begin{align*} g^\prime(x)&=\frac{\cos(x)\cdot x^{7}-\sin(x)\cdot7x^{6}}{(x^{7})^{2}}\\ &=\frac{x^{6}(x\cos(x)-7\sin(x))}{x^{14}}\\ &=\frac{x\cos(x)-7\sin(x)}{x^{8}} \end{align*}$$

\]

Answer:

$g^\prime(x)=\frac{x\cos(x)-7\sin(x)}{x^{8}}$