question\nfind the derivative of the function $f(x)=\frac{5}{x - 1}$.\nprovide your answer…

Explanation:

Step1: Rewrite the function

Rewrite $f(x)=\frac{5}{x - 1}$ as $f(x)=5(x - 1)^{-1}$.

Step2: Apply the power - rule for differentiation

The power - rule states that if $y = ax^n$, then $y^\prime=anx^{n - 1}$. Here $a = 5$ and $n=-1$. Also, by the chain - rule, if $y = u^{-1}$ and $u=x - 1$, then $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. The derivative of $u=x - 1$ with respect to $x$ is $\frac{du}{dx}=1$. $f^\prime(x)=5\times(-1)\times(x - 1)^{-1-1}\times1$.

Step3: Simplify the expression

$f^\prime(x)=-5(x - 1)^{-2}=-\frac{5}{(x - 1)^2}$.

Answer:

$-\frac{5}{(x - 1)^2}$