How do you differentiate $y = \dfrac{{{x^3}}}{{1 - {x^2}}}?$

Hint: According to given in the question we have to differentiate the given expression as in the question $y = \dfrac{{{x^3}}}{{1 - {x^2}}}$. So, first of all to differentiate the given expression we have to use the quotient rule to find the derivation and the formula is as mentioned below:

Formula used: $ \Rightarrow y' = \dfrac{{f'(x)g(x) - f(x)g'(x)}}{{{{\left[ {g(x)} \right]}^2}}}.............(A)$
Where, $f(x)$ is the numerator, $g(x)$ is the denominator, $f'(x)$ is the derivation of the $f(x)$ and same as $g'(x)$ is the derivation of the $g(x)$.
Now, we have to determine the value of $f'(x)$ or we can say that we have to determine the differentiation of $f(x)$
To differentiate the term $f(x)$ we have to use the formula which is as mentioned below:
$ \Rightarrow \dfrac{{d{x^n}}}{{dx}} = n{x^{n - 1}}..................(B)$
Now, we have to determine the value of $g'(x)$ or we can say that we have to determine the differentiation of $g(x)$
Now, we have to substitute the value of $f'(x)$ and $g'(x)$ in the formula (A) as mentioned above to determine the differentiation of the given expression.

Complete step-by-step solution:
Step 1: First of all to differentiate the given expression we have to use the quotient rule to find the derivation and the formula is as mentioned in the solution hint.
Step 2: Now, we have to determine the value of $f'(x)$ or we can say that we have to determine the differentiation of $f(x)$ as mentioned in the solution hint. Hence,
$
   \Rightarrow f'(x) = \dfrac{{df(x)}}{{dx}} \\
   \Rightarrow f'(x) = \dfrac{{d{x^3}}}{{dx}}...............(1)
 $
Step 3: Now, to solve the expression (1) as obtained in the solution step 2 we have to use the formula (B) which is as mentioned in the solution hint. Hence,
$ \Rightarrow f'(x) = 3{x^2}$
Step 4: Now, we have to determine the value of $g'(x)$ or we can say that we have to determine the differentiation of $g(x)$ with the help of the formula which is as mentioned in the solution hint. Hence,
$ \Rightarrow g'(x) = - 2x$
Step 5: Now, we have to substitute the values of $f'(x)$ and $g'(x)$ in the formula (A) which is as mentioned in the solution hint. Hence, on substituting all the values in the formula (A),
$ \Rightarrow y' = \dfrac{{3{x^2}(1 - {x^2}) - ({x^3})(1 - {x^2})}}{{{{(1 - {x^2})}^2}}}............(2)$
Step 6: Now, we have to solve the expression (2) as we already obtained in the solution step 5. Hence,
\[
   \Rightarrow y' = \dfrac{{3{x^2} - 3{x^4} + 2{x^4}}}{{{{(1 - {x^2})}^2}}} \\
   \Rightarrow y' = \dfrac{{ - {x^4} + 3{x^2}}}{{{{(1 - {x^2})}^2}}} \\
   \Rightarrow y' = \dfrac{{{x^2}(3 - {x^2})}}{{{{(1 - {x^2})}^2}}}
 \]

Hence, with the help of the formula (A) and (B) we have determined the differentiation of the given function $y = \dfrac{{{x^3}}}{{1 - {x^2}}} \Rightarrow \dfrac{{ - {x^4} + 3{x^2}}}{{{{(1 - {x^2})}^2}}}$.

Note: It is necessary that we have to determine the differentiation of the functions $f'(x)$ and $g'(x)$ so that we can obtain the differentiation of $f(x)$ and $g(x)$.
We can determine the differentiation of ${x^3}$ with the help of the formula to find the differentiation of ${x^n}$ which is equal to $n{x^{n - 1}}$