How do you differentiate $\dfrac{{x + 1}}{{x - 1}}$?

Hint: According to the question we have to determine the differentiation of the given function $\dfrac{{x + 1}}{{x - 1}}$ as mentioned in the question. So, first of all to determine the differentiation of the given function that the terms are in the form of fraction in which $x + 1$ is the numerator and $x - 1$ is the denominator so, to obtain the differentiation we have to use the quotient rule which is as explained below:
Quotient rule: If the given function is in the form of fraction such as the functions are $f(x)$ and $g(x)$ are in the form of $\dfrac{{f(x)}}{{g(x)}}$ then the quotient are differentiable.
Now, we have to consider our $f(x)$ and $g(x)$ are in the form of $\dfrac{{f(x)}}{{g(x)}}$ which can be determine by the function $\dfrac{{x + 1}}{{x - 1}}$ as mentioned in the question.
Now, to solve the function we have to apply the quotient formula to find the differentiation which is as mentioned below:

Formula used: $ {\left( {\dfrac{f}{g}} \right)'} = \dfrac{{{f'}g - {g'}f}}{{{g^2}}}...............(A)$
Where, f is the numerator of the given function and g is the denominator of the given function.
Now, to find the differentiation we have to use the formula which is as mentioned below:
$ \Rightarrow \dfrac{{dx}}{{dx}} = 1................(B)$
Now, we have to simplify the function obtained after applying formula (B) which by adding and subtracting the terms obtained which can be added and subtracted.

Complete step-by-step solution:
Step 1: First of all to determine the differentiation of the given function that the terms are in the form of fraction in which $x + 1$ is the numerator and $x - 1$ is the denominator so, to obtain the differentiation we have to use the quotient rule which is as explained in the solution hint.
Step 2: Now, we have to consider our $f(x)$ and $g(x)$ are in the form of $\dfrac{{f(x)}}{{g(x)}}$ which can be determine by the function $\dfrac{{x + 1}}{{x - 1}}$ as mentioned in the question. Hence,
$
   \Rightarrow f(x) = x + 1 \\
   \Rightarrow g(x) = x - 1
 $
Step 3: Now, to solve the function we have to apply the quotient formula (A) to find the differentiation which is as mentioned in the solution hint. Hence,
\[\dfrac{d}{{dx}}\left( {\dfrac{{x + 1}}{{x - 1}}} \right) = \dfrac{{(x - 1)\dfrac{d}{{dx}}(x + 1) - (x + 1)\dfrac{d}{{dx}}(x + 1)}}{{{{(x - 1)}^2}}}\]
Step 4: Now, to find the differentiation we have to use the formula (B) which is as mentioned in the solution hint. Hence,
\[ = \dfrac{{(x - 1)(1) - (x + 1)(1)}}{{{{(x - 1)}^2}}}\]
Step 5: Now, we have to simplify the function obtained after applying formula (B) which by adding and subtracting the terms obtained which can be added and subtracted. Hence,
\[
   = \dfrac{{x - 1 - x - 1}}{{{{(x - 1)}^2}}} \\
   = \dfrac{{ - 2}}{{{{(x - 1)}^2}}}
 \]

Hence, with the help of the formula (A) and (B) we have determined the differentiation of the given function which is \[ = \dfrac{{ - 2}}{{{{(x - 1)}^2}}}\].

Note: In the given function as we can see that the function is in the form of fraction as $\dfrac{{f(x)}}{{g(x)}}$ hence, to determine the differentiation it is necessary that we have to use the quotient rule.
According to the quotient rule we have to take the denominator as constant term and differentiate the numerator and then we take the numerator as constant term and differentiate the denominator and then we have to subtract both and divide by the square of the denominator.