Answer

Verified

338.1k+ views

**Hint:**

Here, we have to find the derivative of the given function. We will use the derivative formula to find the derivative of the logarithmic function. Then we will find the derivative of the algebraic function by using the concept of Implicit differentiation. We will simplify the equation further to get the required answer.

**Formula Used:**

We will use the following formulas:

1) Derivative formula: \[\dfrac{d}{{dx}}\left( {\ln x} \right) = \dfrac{1}{x}\]

2) Derivative formula: \[\dfrac{d}{{dx}}\left( {{x^n}} \right) = n{x^{n - 1}}\]

**Complete step by step solution:**

We are given with a function \[y = \ln \left( {8{x^2} + 9{y^2}} \right)\]

Now, we will find the derivative of the given function.

Now, we will find the derivative of the logarithmic function followed by the derivative of the algebraic function simultaneously.

Using the derivative formula \[\dfrac{d}{{dx}}\left( {\ln x} \right) = \dfrac{1}{x}\], we get

\[ \Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{1}{{8{x^2} + 9{y^2}}}\left[ {\dfrac{d}{{dx}}\left( {8{x^2} + 9{y^2}} \right)} \right]\]

Simplifying the equation, we get

\[ \Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{1}{{8{x^2} + 9{y^2}}}\dfrac{d}{{dx}}\left( {8{x^2}} \right) + \dfrac{1}{{8{x^2} + 9{y^2}}}\dfrac{d}{{dx}}\left( {9{y^2}} \right)\]

Now, by using the derivative formula \[\dfrac{d}{{dx}}\left( {{x^n}} \right) = n{x^{n - 1}}\], we get

\[ \Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{{8 \cdot 2x}}{{8{x^2} + 9{y^2}}} + \dfrac{{9 \cdot 2y}}{{8{x^2} + 9{y^2}}}\dfrac{{dy}}{{dx}}\]

Multiplying the terms, we get

\[ \Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{{16x}}{{8{x^2} + 9{y^2}}} + \dfrac{{18y}}{{8{x^2} + 9{y^2}}}\dfrac{{dy}}{{dx}}\]

Rewriting the equation, we get

\[ \Rightarrow \dfrac{{dy}}{{dx}} - \dfrac{{18y}}{{8{x^2} + 9{y^2}}}\dfrac{{dy}}{{dx}} = \dfrac{{16x}}{{8{x^2} + 9{y^2}}}\]

Now, by taking out the common factor, we get

\[ \Rightarrow \dfrac{{dy}}{{dx}}\left( {1 - \dfrac{{18y}}{{8{x^2} + 9{y^2}}}} \right) = \dfrac{{16x}}{{8{x^2} + 9{y^2}}}\]

Taking LCM of the terms inside the bracket on the RHS, we get

\[ \Rightarrow \dfrac{{dy}}{{dx}}\left( {1 \times \dfrac{{8{x^2} + 9{y^2}}}{{8{x^2} + 9{y^2}}} - \dfrac{{18y}}{{8{x^2} + 9{y^2}}}} \right) = \dfrac{{16x}}{{8{x^2} + 9{y^2}}}\]

\[ \Rightarrow \dfrac{{dy}}{{dx}}\left( {\dfrac{{8{x^2} + 9{y^2} - 18y}}{{8{x^2} + 9{y^2}}}} \right) = \dfrac{{16x}}{{8{x^2} + 9{y^2}}}\]

Now, by rewriting the terms, we get

\[ \Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{{\dfrac{{16x}}{{8{x^2} + 9{y^2}}}}}{{\left( {\dfrac{{8{x^2} + 9{y^2} - 18y}}{{8{x^2} + 9{y^2}}}} \right)}}\]

Cancelling out the same terms of the fractions, we get

\[ \Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{{16x}}{{8{x^2} + 9{y^2} - 18y}}\]

**Therefore, the derivative \[\dfrac{{dy}}{{dx}}\] of the function \[y = \ln \left( {8{x^2} + 9{y^2}} \right)\] is \[\dfrac{{16x}}{{8{x^2} + 9{y^2} - 18y}}\].**

**Note:**

We know that Differentiation is a method of finding the derivative of a function and finding the rate of change of function with respect to one variable. But here, we are using the concept of Implicit differentiation. Implicit Differentiation is a process of finding the derivative of a function when the function has both the terms\[x\] and\[y\]. Implicit Differentiation is similar to the process of differentiation and uses the same formula used for differentiation.

Recently Updated Pages

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

How do you arrange NH4 + BF3 H2O C2H2 in increasing class 11 chemistry CBSE

Is H mCT and q mCT the same thing If so which is more class 11 chemistry CBSE

What are the possible quantum number for the last outermost class 11 chemistry CBSE

Is C2 paramagnetic or diamagnetic class 11 chemistry CBSE

What happens when entropy reaches maximum class 11 chemistry JEE_Main

Trending doubts

How do you solve x2 11x + 28 0 using the quadratic class 10 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

Difference Between Plant Cell and Animal Cell

Which places in India experience sunrise first and class 9 social science CBSE

The list which includes subjects of national importance class 10 social science CBSE

What is pollution? How many types of pollution? Define it

State the laws of reflection of light