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

Find the derivative of the following:
y2−3xy+x2=x{{y}^{2}}-3xy+{{x}^{2}}=x

Explanation

Solution

Hint : If u and v are functions in terms of x then their differentiation with respect to x is given byddx(uv)=u×dvdx+v×dudx\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 ddx(uv)=u×dvdx+v×dudx\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
ddx(y2)−3ddx(xy)+ddx(x2)=ddx(x)\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 un{{u}^{n}} in terms of u is given by ddu(un)=nun−1\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.
2ydydx−3(x×dydx+y)+2x=12y\dfrac{dy}{dx}-3\left( x\times \dfrac{dy}{dx}+y \right)+2x=1
Simplifying the equation we get,
2ydydx−3xdydx+3y+2x=12y\dfrac{dy}{dx}-3x\dfrac{dy}{dx}+3y+2x=1
Subtracting with 3y and 2x on both sides we get,
2ydydx−3xdydx=1−3y−2x2y\dfrac{dy}{dx}-3x\dfrac{dy}{dx}=1-3y-2x
Taking dydx\dfrac{dy}{dx} as common in the LHS we get,
dydx(2y−3x)=1−3y−2x\dfrac{dy}{dx}\left( 2y-3x \right)=1-3y-2x
On dividing with (2y - 3x) on both sides we get,
dydx=1−3y−2x2y−3x\dfrac{dy}{dx}=\dfrac{1-3y-2x}{2y-3x}
Hence, we get the derivative of the equation y2−3xy+x2=x{{y}^{2}}-3xy+{{x}^{2}}=x as dydx=1−3y−2x2y−3x\dfrac{dy}{dx}=\dfrac{1-3y-2x}{2y-3x}.

Note : The formula for derivative of un{{u}^{n}} in terms of u is given by ddu(un)=nun−1\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 dydx\dfrac{dy}{dx}. Using previously mentioned rules carefully complete the basic mathematical operations like addition, subtraction to calculate the final answer.