Question

Find the derivative of the following:
At \[\left( 1,-1 \right)\]find \[{y}'\]for \[{{x}^{4}}+4{{x}^{2}}{{y}^{3}}+{{y}^{2}}=2y\]

Answer 1

Hint:If u and v are two differentiable functions of x then \[\dfrac{d}{dx}\left( uv \right)=u\times \dfrac{dv}{dx}+v\times \dfrac{du}{dx}\]and this formula is called product rule. In this problem u and v are two differentiable functions of x so apply the product rule. If they ask to find the derivative at a particular point substitute the values of x and y in the obtained derivative value.

Complete step-by-step answer:
Given that \[{{x}^{4}}+4{{x}^{2}}{{y}^{3}}+{{y}^{2}}=2y\]
First apply derivative on both sides with respect to x then we will get
\[4{{x}^{3}}+4\left( {{y}^{3}}\times 2x+{{x}^{2}}3{{y}^{2}}\dfrac{dy}{dx} \right)+\left( 2y \right)\dfrac{dy}{dx}=2\dfrac{dy}{dx}\]. . . . . . . . . . . . . . . . . . (1)
\[4{{x}^{3}}+8x{{y}^{3}}=\dfrac{dy}{dx}\left( -12{{x}^{2}}{{y}^{2}}-2y+2 \right)\]. . . . . . . . . . . . . . . . . . . . (2)
\[\dfrac{dy}{dx}=\dfrac{4{{x}^{3}}+8x{{y}^{3}}}{-12{{x}^{2}}{{y}^{2}}-2y+2}\]. . . . . . . . . . . . . . . . . . (3)
The derivative of the following \[{y}'\]is \[\dfrac{4{{x}^{3}}+8x{{y}^{3}}}{-12{{x}^{2}}{{y}^{2}}-2y+2}\]
Given they asked us to find the derivative at the point \[(1,-1)\]
\[\dfrac{dy}{dx}=\dfrac{4{{\left( 1 \right)}^{3}}+8\left( 1 \right){{\left( -1 \right)}^{3}}}{-12{{\left( 1 \right)}^{2}}{{\left( -1 \right)}^{2}}-2\left( -1 \right)+2}\]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4)
\[\dfrac{dy}{dx}=\dfrac{-4}{-12+2+2}\]
\[\dfrac{dy}{dx}=\dfrac{-4}{-8}\]
\[\dfrac{dy}{dx}=\dfrac{1}{2}\]
So, the derivative at the point \[(1,-1)\]is \[\dfrac{dy}{dx}=\dfrac{1}{2}\]

Note: The formula for derivative of \[{{u}^{n}}=n{{u}^{n-1}}\]where n is any integer and u is any variable like x or y. the x component with respect x is 1 and derivative of y component with respect to x is \[\dfrac{dy}{dx}\]as shown this is used in the above problem. Carefully do the basic mathematical operations like addition, subtraction then we will get the required answer.