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# Find the derivative of the following: ${{y}^{2}}-3xy+{{x}^{2}}=x$

Last updated date: 13th Jun 2024
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Hint: If u and v are functions in terms of x then their differentiation with respect to x is given by$\dfrac{d}{dx}\left( uv \right)=u\times \dfrac{dv}{dx}+v\times \dfrac{du}{dx}$and this formula is called as the product rule. Use the product rule to simplify the given equation and calculate the derivative of the equation.

We know the differentiation with respect to x for two functions u and v in terms of x is given by $\dfrac{d}{dx}\left( uv \right)=u\times \dfrac{dv}{dx}+v\times \dfrac{du}{dx}$.
Applying the above mentioned formula to simplify the given equation.
First apply derivative on both sides of the equation with respect to x then we will get
$\dfrac{d}{dx}\left( {{y}^{2}} \right)-3\dfrac{d}{dx}\left( xy \right)+\dfrac{d}{dx}\left( {{x}^{2}} \right)=\dfrac{d}{dx}\left( x \right)$
Using the formula for derivative of ${{u}^{n}}$ in terms of u is given by $\dfrac{d}{du}\left( {{u}^{n}} \right)=n{{u}^{n-1}}$ where n is any integer and u is any variable we get,
We know the differentiation of x with respect to x is 1.
$2y\dfrac{dy}{dx}-3\left( x\times \dfrac{dy}{dx}+y \right)+2x=1$
Simplifying the equation we get,
$2y\dfrac{dy}{dx}-3x\dfrac{dy}{dx}+3y+2x=1$
Subtracting with 3y and 2x on both sides we get,
$2y\dfrac{dy}{dx}-3x\dfrac{dy}{dx}=1-3y-2x$
Taking $\dfrac{dy}{dx}$ as common in the LHS we get,
$\dfrac{dy}{dx}\left( 2y-3x \right)=1-3y-2x$
On dividing with (2y - 3x) on both sides we get,
$\dfrac{dy}{dx}=\dfrac{1-3y-2x}{2y-3x}$
Hence, we get the derivative of the equation ${{y}^{2}}-3xy+{{x}^{2}}=x$ as $\dfrac{dy}{dx}=\dfrac{1-3y-2x}{2y-3x}$.

Note: The formula for derivative of ${{u}^{n}}$ in terms of u is given by $\dfrac{d}{du}\left( {{u}^{n}} \right)=n{{u}^{n-1}}$ where n is any integer and u is any variable. The derivative of x component with respect to x is 1 and derivative of y component with respect to x is $\dfrac{dy}{dx}$. Using previously mentioned rules carefully complete the basic mathematical operations like addition, subtraction to calculate the final answer.