How do you find $\dfrac{{dy}}{{dx}}$ given $y + {y^3} = {x^2}$?

Hint: In the given question, we are required to find the derivative of the function provided to us. So, we will differentiate both sides of the equation given to us with respect to x using the power rule of differentiation. Then, we will take the $\dfrac{{dy}}{{dx}}$ term common from the left side of the equation and find its value using the method of transposition.

Complete step by step answer:
We are given the equation $y + {y^3} = {x^2}$ and we have to find the value of $\dfrac{{dy}}{{dx}}$.
So, we differentiate both sides of the equation with respect to x.
$ \Rightarrow \dfrac{d}{{dx}}\left( {y + {y^3}} \right) = \dfrac{d}{{dx}}\left( {{x^2}} \right)$
Now, we know that $\dfrac{d}{{dx}}\left( {a + b} \right) = \dfrac{d}{{dx}}\left( a \right) + \dfrac{d}{{dx}}\left( b \right)$.
So, we get,
$ \Rightarrow \dfrac{d}{{dx}}\left( y \right) + \dfrac{d}{{dx}}\left( {{y^3}} \right) = \dfrac{d}{{dx}}\left( {{x^2}} \right)$

Using the power rule of differentiation $\dfrac{{d\left( {{x^n}} \right)}}{{dx}} = n{x^{n - 1}}$, we get,
$ \Rightarrow \dfrac{d}{{dx}}\left( y \right) + \dfrac{d}{{dx}}\left( {{y^3}} \right) = 2\left( {{x^{2 - 1}}} \right)$
Now, we will follow the chain rule of differentiation $\dfrac{{d\left( {fog\left( x \right)} \right)}}{{dx}} = f'\left( {g\left( x \right)} \right)g'\left( x \right)$.
$ \Rightarrow \dfrac{{dy}}{{dx}} + \left( {3{y^{3 - 1}}} \right)\dfrac{{dy}}{{dx}} = 2\left( {{x^{2 - 1}}} \right)$
Simplifying the expression,
$ \Rightarrow \dfrac{{dy}}{{dx}} + 3{y^2}\dfrac{{dy}}{{dx}} = 2x$
Taking $\dfrac{{dy}}{{dx}}$ common from the left side of the equation, we get,
$ \Rightarrow \dfrac{{dy}}{{dx}}\left( {1 + 3{y^2}} \right) = 2x$
Dividing both the sides of the equation by $\left( {1 + 3{y^2}} \right)$, we get,
$ \therefore \dfrac{{dy}}{{dx}} = \dfrac{{2x}}{{\left( {1 + 3{y^2}} \right)}}$

So, the value of $\dfrac{{dy}}{{dx}}$ is $\dfrac{{2x}}{{\left( {1 + 3{y^2}} \right)}}$.

Note: Chain rule of differentiation $\dfrac{{d\left( {fog\left( x \right)} \right)}}{{dx}} = f'\left( {g\left( x \right)} \right)g'\left( x \right)$ helps us to differentiate the composite and complex functions layer by layer. Power rule of differentiation helps to differentiate the power function as ${x^n}$ with respect to x. Transposition rule states that both sides of the equation remain equal if we multiply, divide, add or subtract the same quantity on both sides.