Find \[$ Y^{\prime} \$\] And \[$ Y^{\prime \prime} \$\] For The Equation \[$ X^2 + 6xy - 8y^2 = 7 \$\].\[$ Y^{\prime} = \$\]\[$ Y^{\prime \prime} = \$\]

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Introduction

Implicit differentiation is a powerful technique used to find the derivatives of implicitly defined functions. In this article, we will explore how to use implicit differentiation to find the first and second derivatives of the given equation { x^2 + 6xy - 8y^2 = 7 $}$. This technique is essential in calculus and has numerous applications in various fields, including physics, engineering, and economics.

Implicit Differentiation

Implicit differentiation is a method used to find the derivative of an implicitly defined function. It involves differentiating both sides of the equation with respect to the independent variable, while treating the dependent variable as a function of the independent variable.

Step 1: Differentiate Both Sides of the Equation

To find the derivative of the given equation, we will differentiate both sides of the equation with respect to x. We will use the product rule and the chain rule to differentiate the terms involving y.

{ \frac{d}{dx} (x^2 + 6xy - 8y^2) = \frac{d}{dx} (7) $}$

Using the product rule, we get:

{ 2x + 6y + 6xy' - 16yy' = 0 $}$

Step 2: Solve for y'

Now, we will solve for y' by isolating it on one side of the equation.

{ 6xy' - 16yy' = -2x - 6y $}$

Factoring out y', we get:

{ y'(6x - 16y) = -2x - 6y $}$

Dividing both sides by (6x - 16y), we get:

{ y' = \frac{-2x - 6y}{6x - 16y} $}$

Step 3: Find the Second Derivative

To find the second derivative, we will differentiate the first derivative with respect to x.

{ \frac{d}{dx} \left( \frac{-2x - 6y}{6x - 16y} \right) $}$

Using the quotient rule, we get:

{ \frac{(6x - 16y)(-2) - (-2x - 6y)(6 - 16y')}{(6x - 16y)^2} $}$

Simplifying the expression, we get:

{ \frac{-12x + 32y + 12x + 36y - 96y^2}{(6x - 16y)^2} $}$

Combining like terms, we get:

{ \frac{68y - 96y^2}{(6x - 16y)^2} $}$

Conclusion

In this article, we used implicit differentiation to find the first and second derivatives of the given equation { x^2 + 6xy - 8y^2 = 7 $}$. The first derivative is { y' = \frac{-2x - 6y}{6x - 16y} $}$, and the second derivative is { \frac{68y - 96y^2}{(6x - 16y)^2} $}$. Implicit differentiation is a powerful technique that can be used to find the derivatives of implicitly defined functions.

Applications of Implicit Differentiation

Implicit differentiation has numerous applications in various fields, including physics, engineering, and economics. Some of the applications of implicit differentiation include:

  • Physics: Implicit differentiation is used to find the velocity and acceleration of an object in terms of its position and time.
  • Engineering: Implicit differentiation is used to find the stress and strain of a material in terms of its displacement and time.
  • Economics: Implicit differentiation is used to find the demand and supply curves of a product in terms of its price and quantity.

Example Problems

Here are some example problems that illustrate the use of implicit differentiation:

Example 1:

Find the derivative of the equation { x^2 + 2xy - 3y^2 = 5 $}$.

Using implicit differentiation, we get:

{ 2x + 2y + 2xy' - 6yy' = 0 $}$

Solving for y', we get:

{ y' = \frac{-2x - 2y}{2x - 6y} $}$

Example 2:

Find the second derivative of the equation { x^2 + 3xy - 2y^2 = 4 $}$.

Using implicit differentiation, we get:

{ 2x + 3y + 3xy' - 4yy' = 0 $}$

Solving for y', we get:

{ y' = \frac{-2x - 3y}{2x - 4y} $}$

Differentiating the first derivative with respect to x, we get:

{ \frac{d}{dx} \left( \frac{-2x - 3y}{2x - 4y} \right) $}$

Using the quotient rule, we get:

{ \frac{(2x - 4y)(-2) - (-2x - 3y)(2 - 8y')}{(2x - 4y)^2} $}$

Simplifying the expression, we get:

{ \frac{-4x + 8y + 4x + 6y - 24y^2}{(2x - 4y)^2} $}$

Combining like terms, we get:

{ \frac{14y - 24y^2}{(2x - 4y)^2} $}$

Conclusion

In this article, we used implicit differentiation to find the first and second derivatives of two given equations. The first derivative is { y' = \frac{-2x - 2y}{2x - 6y} $}$ and { y' = \frac{-2x - 3y}{2x - 4y} $}$, and the second derivative is { \frac{14y - 24y^2}{(2x - 4y)^2} $}$. Implicit differentiation is a powerful technique that can be used to find the derivatives of implicitly defined functions.

Final Thoughts

Q: What is implicit differentiation?

A: Implicit differentiation is a technique used to find the derivatives of implicitly defined functions. It involves differentiating both sides of the equation with respect to the independent variable, while treating the dependent variable as a function of the independent variable.

Q: How do I use implicit differentiation to find the derivative of an implicitly defined function?

A: To use implicit differentiation, follow these steps:

  1. Differentiate both sides of the equation with respect to the independent variable.
  2. Use the product rule and the chain rule to differentiate the terms involving the dependent variable.
  3. Solve for the derivative of the dependent variable.

Q: What are some common mistakes to avoid when using implicit differentiation?

A: Some common mistakes to avoid when using implicit differentiation include:

  • Forgetting to differentiate both sides of the equation.
  • Failing to use the product rule and the chain rule when differentiating terms involving the dependent variable.
  • Not solving for the derivative of the dependent variable.

Q: Can I use implicit differentiation to find the second derivative of an implicitly defined function?

A: Yes, you can use implicit differentiation to find the second derivative of an implicitly defined function. To do this, differentiate the first derivative with respect to the independent variable.

Q: What are some real-world applications of implicit differentiation?

A: Some real-world applications of implicit differentiation include:

  • Finding the velocity and acceleration of an object in terms of its position and time.
  • Finding the stress and strain of a material in terms of its displacement and time.
  • Finding the demand and supply curves of a product in terms of its price and quantity.

Q: How do I choose the correct method for finding the derivative of an implicitly defined function?

A: To choose the correct method for finding the derivative of an implicitly defined function, consider the following:

  • If the function is explicitly defined, use the power rule and the sum rule to find the derivative.
  • If the function is implicitly defined, use implicit differentiation to find the derivative.
  • If the function is a combination of explicitly and implicitly defined functions, use a combination of the power rule, the sum rule, and implicit differentiation to find the derivative.

Q: Can I use implicit differentiation to find the derivative of a function that is not implicitly defined?

A: No, you cannot use implicit differentiation to find the derivative of a function that is not implicitly defined. Implicit differentiation is a technique used to find the derivatives of implicitly defined functions.

Q: What are some common types of implicitly defined functions?

A: Some common types of implicitly defined functions include:

  • Functions of the form { x^2 + y^2 = r^2 $}$, where r is a constant.
  • Functions of the form { x^2 - y^2 = r^2 $}$, where r is a constant.
  • Functions of the form { xy = r $}$, where r is a constant.

Q: How do I use implicit differentiation to find the derivative of a function of the form { x^2 + y^2 = r^2 $}$?

A: To use implicit differentiation to find the derivative of a function of the form { x^2 + y^2 = r^2 $}$, follow these steps:

  1. Differentiate both sides of the equation with respect to x.
  2. Use the chain rule to differentiate the term involving y.
  3. Solve for the derivative of y.

Q: What are some common applications of implicit differentiation in physics?

A: Some common applications of implicit differentiation in physics include:

  • Finding the velocity and acceleration of an object in terms of its position and time.
  • Finding the force and energy of an object in terms of its position and time.
  • Finding the momentum and kinetic energy of an object in terms of its position and time.

Q: How do I use implicit differentiation to find the derivative of a function of the form { xy = r $}$?

A: To use implicit differentiation to find the derivative of a function of the form { xy = r $}$, follow these steps:

  1. Differentiate both sides of the equation with respect to x.
  2. Use the product rule to differentiate the term involving y.
  3. Solve for the derivative of y.

Q: What are some common applications of implicit differentiation in engineering?

A: Some common applications of implicit differentiation in engineering include:

  • Finding the stress and strain of a material in terms of its displacement and time.
  • Finding the force and energy of a system in terms of its displacement and time.
  • Finding the momentum and kinetic energy of a system in terms of its displacement and time.

Q: How do I use implicit differentiation to find the derivative of a function of the form { x^2 - y^2 = r^2 $}$?

A: To use implicit differentiation to find the derivative of a function of the form { x^2 - y^2 = r^2 $}$, follow these steps:

  1. Differentiate both sides of the equation with respect to x.
  2. Use the chain rule to differentiate the term involving y.
  3. Solve for the derivative of y.

Conclusion

Implicit differentiation is a powerful technique used to find the derivatives of implicitly defined functions. It has numerous applications in various fields, including physics, engineering, and economics. By using implicit differentiation, we can find the velocity and acceleration of an object in terms of its position and time, the stress and strain of a material in terms of its displacement and time, and the demand and supply curves of a product in terms of its price and quantity.