# Find \[\dfrac{{dy}}{{dx}}\] of \[{x^2} + xy + {y^2} = 100\].

Last updated date: 18th Mar 2023

•

Total views: 307.2k

•

Views today: 6.87k

Answer

Verified

307.2k+ views

Hint:- Use the product rule to find \[\dfrac{{dy}}{{dx}}\] of \[xy\] the derivative.

Given equation in the question is ,

\[ \Rightarrow {x^2} + xy + {y^2} = 100\] (1)

We had to find the \[\dfrac{{dy}}{{dx}}\] of the given equation 1. So, for that,

We must find the derivative of the given equation 1 with respect to x.

Finding \[\dfrac{{dy}}{{dx}}\] of the given equation,

\[ \Rightarrow 2x + \left( {y + x\dfrac{{dy}}{{dx}}({\text{By applying product rule)}}} \right) + 2y\dfrac{{dy}}{{dx}} = 0\] (2)

Now solving equation 2.

Taking \[\dfrac{{dy}}{{dx}}\] common from equation 2. It becomes,

\[ \Rightarrow 2x + y + \dfrac{{dy}}{{dx}}(x + 2y) = 0\]

Now taking \[(2x + y)\] to the RHS of the above equation. It becomes,

\[ \Rightarrow \dfrac{{dy}}{{dx}}(x + 2y) = - \left( {2x + y} \right)\]

Now, dividing both sides of the above equation by \[(x + 2y)\]. We get,

\[ \Rightarrow \dfrac{{dy}}{{dx}} = - \dfrac{{\left( {2x + y} \right)}}{{(x + 2y)}}\]

Hence, value of \[\dfrac{{dy}}{{dx}}\] for the given equation will be \[ - \dfrac{{\left( {2x + y} \right)}}{{(x + 2y)}}\].

Note:- Whenever we came up with this type of problem then first, find derivative of given

equation with respect to x, using different derivative formulas and various rules like product

rule, quotient rule and chain rule etc. Then take all the terms with \[\dfrac{{dy}}{{dx}}\] to one side of the equation.

As this will be the easiest and efficient way to find the value of \[\dfrac{{dy}}{{dx}}\] for the given equation.

Given equation in the question is ,

\[ \Rightarrow {x^2} + xy + {y^2} = 100\] (1)

We had to find the \[\dfrac{{dy}}{{dx}}\] of the given equation 1. So, for that,

We must find the derivative of the given equation 1 with respect to x.

Finding \[\dfrac{{dy}}{{dx}}\] of the given equation,

\[ \Rightarrow 2x + \left( {y + x\dfrac{{dy}}{{dx}}({\text{By applying product rule)}}} \right) + 2y\dfrac{{dy}}{{dx}} = 0\] (2)

Now solving equation 2.

Taking \[\dfrac{{dy}}{{dx}}\] common from equation 2. It becomes,

\[ \Rightarrow 2x + y + \dfrac{{dy}}{{dx}}(x + 2y) = 0\]

Now taking \[(2x + y)\] to the RHS of the above equation. It becomes,

\[ \Rightarrow \dfrac{{dy}}{{dx}}(x + 2y) = - \left( {2x + y} \right)\]

Now, dividing both sides of the above equation by \[(x + 2y)\]. We get,

\[ \Rightarrow \dfrac{{dy}}{{dx}} = - \dfrac{{\left( {2x + y} \right)}}{{(x + 2y)}}\]

Hence, value of \[\dfrac{{dy}}{{dx}}\] for the given equation will be \[ - \dfrac{{\left( {2x + y} \right)}}{{(x + 2y)}}\].

Note:- Whenever we came up with this type of problem then first, find derivative of given

equation with respect to x, using different derivative formulas and various rules like product

rule, quotient rule and chain rule etc. Then take all the terms with \[\dfrac{{dy}}{{dx}}\] to one side of the equation.

As this will be the easiest and efficient way to find the value of \[\dfrac{{dy}}{{dx}}\] for the given equation.

Recently Updated Pages

Calculate the entropy change involved in the conversion class 11 chemistry JEE_Main

The law formulated by Dr Nernst is A First law of thermodynamics class 11 chemistry JEE_Main

For the reaction at rm0rm0rmC and normal pressure A class 11 chemistry JEE_Main

An engine operating between rm15rm0rm0rmCand rm2rm5rm0rmC class 11 chemistry JEE_Main

For the reaction rm2Clg to rmCrmlrm2rmg the signs of class 11 chemistry JEE_Main

The enthalpy change for the transition of liquid water class 11 chemistry JEE_Main

Trending doubts

Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Write the 6 fundamental rights of India and explain in detail

Write a letter to the principal requesting him to grant class 10 english CBSE

List out three methods of soil conservation

Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE

Write a letter to the Principal of your school to plead class 10 english CBSE