Answer

Verified

452.4k+ 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

How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE

Why Are Noble Gases NonReactive class 11 chemistry CBSE

Let X and Y be the sets of all positive divisors of class 11 maths CBSE

Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE

Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE

Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE

Trending doubts

Which are the Top 10 Largest Countries of the World?

How many crores make 10 million class 7 maths CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Fill the blanks with proper collective nouns 1 A of class 10 english CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

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

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths