Question

How do you differentiate $2xy+{{y}^{2}}=x+y$?

Answer 1

Hint: Derivative is the rate of change of a function. To find the derivative of a function, we have to differentiate it with respect to $x$. Finding out the derivative of a function implies that we are finding out the slope of the function.$\dfrac{dy}{dx}$ tells us about the slope of a curve. In trigonometric terms, $\tan \theta $ , where $\theta $ is the inclination of a line with the positive $x$-axis measured in anti-clockwise direction , tells us about the slope of a line.

Complete step by step solution:
Here we have $2xy+{{y}^{2}}=x+y$.
Now let us differentiate this with respect to $x$.
Before doing that, we should know the following :
$\Rightarrow \dfrac{d\left( {{x}^{n}} \right)}{dx}=n{{x}^{n-1}}$ .
And also we need to make use of the $uv$ rule of differentiation. Whenever we have two functions namely, $h\left( x \right),g\left( x \right)$ in this form $h\left( x \right)g\left( x \right)$ , then we differentiate such a function using this rule. It states the following :
$\Rightarrow \dfrac{d}{dx}\left( h\left( x \right)g\left( x \right) \right)=g\left( x \right)\dfrac{d}{dx}\left( h\left( x \right) \right)+h\left( x \right)\dfrac{d}{dx}\left( g\left( x \right) \right)$ .
Let us use this piece of information while differentiating.
Now, let us differentiate.
Upon differentiating, we get the following :
$\begin{align}
  & \Rightarrow 2xy+{{y}^{2}}=x+y \\
 & \Rightarrow 2y+2x\dfrac{dy}{dx}+2y\dfrac{dy}{dx}=1+\dfrac{dy}{dx} \\
\end{align}$
Let us gather all the $\dfrac{dy}{dx}$ onto one side of the equation.
Upon doing so, we get the following :
$\begin{align}
  & \Rightarrow 2xy+{{y}^{2}}=x+y \\
 & \Rightarrow 2y+2x\dfrac{dy}{dx}+2y\dfrac{dy}{dx}=1+\dfrac{dy}{dx} \\
 & \Rightarrow \dfrac{dy}{dx}\left( 2x+2y-1 \right)+2y=1 \\
\end{align}$
Let us send the $2y$onto the right hand side of the equation from the left hand side.
Upon doing so, we get the following :
$\begin{align}
  & \Rightarrow 2xy+{{y}^{2}}=x+y \\
 & \Rightarrow 2y+2x\dfrac{dy}{dx}+2y\dfrac{dy}{dx}=1+\dfrac{dy}{dx} \\
 & \Rightarrow \dfrac{dy}{dx}\left( 2x+2y-1 \right)+2y=1 \\
 & \Rightarrow \dfrac{dy}{dx}\left( 2x+2y-1 \right)=1-2y \\
\end{align}$
Let us bring the $2x+2y-1$ onto the left hand side of the equation.
Upon doing so, we get the following :
$\Rightarrow \dfrac{dy}{dx}=\dfrac{1-2y}{2x+2y-1}$.

$\therefore $ Hence , the derivative of the function $2xy+{{y}^{2}}=x+y$ is $\dfrac{1-2y}{2x+2y-1}$.

Note: We have to be careful while applying the formula. We should carefully compare and find out what $h\left( x \right)\And g\left( x \right)$ are . It is advisable to learn the derivatives of all the standard functions in order to solve the question quickly. We should also learn all the rules of differentiation.