Question

Find the derivative of $y = \dfrac{x}{{x - 1}}$ .

Answer 1

Hint: We will use quotient rule to find the derivative. According to the quotient rule, if two functions, say f(x) and g(x), the derivative of $\dfrac{{f(x)}}{{g(x)}}$ will be \[\dfrac{d}{{dx}}\left( {\dfrac{{f\left( x \right)}}{{g\left( x \right)}}} \right) = \dfrac{{g\left( x \right)\dfrac{{df\left( x \right)}}{{dx}} - f\left( x \right)\dfrac{{dg\left( x \right)}}{{dx}}}}{{{{\left( {g\left( x \right)} \right)}^2}}}\] . Furthermore, differentiate the functions f(x) and g(x) separately. As a result, if the answer is reducible, it should be simplified.

Complete step by step answer:
We have to find the derivative of $y = \dfrac{x}{{x - 1}}$
 We will assume $f(x) = x$ and $g(x) = x - 1$ . then, we will use quotient rule
\[ \Rightarrow \dfrac{d}{{dx}}\left( {\dfrac{{f\left( x \right)}}{{g\left( x \right)}}} \right) = \dfrac{{g\left( x \right)\dfrac{{df\left( x \right)}}{{dx}} - f\left( x \right)\dfrac{{dg\left( x \right)}}{{dx}}}}{{{{\left( {g\left( x \right)} \right)}^2}}}\] (1)
Here, we don’t know the value of \[\dfrac{d}{{dx}}f(x)\] and \[\dfrac{d}{{dx}}g(x)\]
We will first find the value of \[\dfrac{d}{{dx}}f(x)\] and \[\dfrac{d}{{dx}}g(x)\]
So, using the formula \[\dfrac{d}{{dx}}\left( {{x^n}} \right) = n{x^{n - 1}}\] ,we will find the derivative
\[ \Rightarrow \dfrac{d}{{dx}}f(x) = \dfrac{d}{{dx}}x = 1\]
\[ \Rightarrow \dfrac{d}{{dx}}g(x) = \dfrac{d}{{dx}}(x - 1) = 1\]
We will now put the value in the equation 1
\[ \Rightarrow \dfrac{d}{{dx}}\left( {\dfrac{{f\left( x \right)}}{{g\left( x \right)}}} \right) = \dfrac{{g\left( x \right)\dfrac{{df\left( x \right)}}{{dx}} - f\left( x \right)\dfrac{{dg\left( x \right)}}{{dx}}}}{{{{\left( {g\left( x \right)} \right)}^2}}}\]
\[ \Rightarrow \dfrac{d}{{dx}}\left( {\dfrac{x}{{x - 1}}} \right) = \dfrac{{(x - 1)1 - x1}}{{{{\left( {x - 1} \right)}^2}}}\]
\[ \Rightarrow \dfrac{d}{{dx}}\left( {\dfrac{x}{{x - 1}}} \right) = \dfrac{{ - 1}}{{{{\left( {x - 1} \right)}^2}}}\]
Hence, the derivative of $y = \dfrac{x}{{x - 1}}$ is \[\dfrac{{ - 1}}{{{{\left( {x - 1} \right)}^2}}}\] .

Note:
When two functions are in division form, the quotient rule of differentiation should always be used. Because there is a function in the denominator, there is no other method to solve this derivative. Keep in mind that attempting to simplify the function such that it is no longer in fraction form will just make it more complicated.