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

Question

Vedantu Content Team · Accepted Answer

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.Complete step-by-step answer: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.

Find the derivative of the following:
\[{{y}^{2}}-3xy+{{x}^{2}}=x\]

NCERT Book Solutions

Reference Book Solutions

Question Papers

Exams

Quick Links

Find the derivative of the following: \[{{y}^{2}}-3xy+{{x}^{2}}=x\]

Book your FREE Online counselling session

Your FREE Online counselling session has been booked!

Please try again later