Find $y^{\prime}$ And $y^{\prime \prime}$ For $ X 2 + 4 X Y − 3 Y 2 = 6 X^2 + 4xy - 3y^2 = 6 X 2 + 4 X Y − 3 Y 2 = 6 [/tex].$y^{\prime} =$$y^{\prime \prime} =$

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 function $x^2 + 4xy - 3y^2 = 6$. 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 function, we will differentiate both sides of the equation with respect to $x$. We will treat $y$ as a function of $x$ and use the chain rule to differentiate the terms involving $y$.

$$\frac{d}{dx}(x^2 + 4xy - 3y^2) = \frac{d}{dx}(6)$$

Using the power rule and the product rule, we get:

$$2x + 4y + 4x\frac{dy}{dx} - 6y\frac{dy}{dx} = 0$$

Step 2: Solve for $\frac{dy}{dx}$

Now, we will solve for $\frac{dy}{dx}$ by isolating it on one side of the equation.

$$4x\frac{dy}{dx} - 6y\frac{dy}{dx} = -2x - 4y$$

Factoring out $\frac{dy}{dx}$, we get:

$$(4x - 6y)\frac{dy}{dx} = -2x - 4y$$

Dividing both sides by $(4x - 6y)$, we get:

$$\frac{dy}{dx} = \frac{-2x - 4y}{4x - 6y}$$

Step 3: Find the Second Derivative

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

$$\frac{d^2y}{dx^2} = \frac{(4x - 6y)(-2) - (-2x - 4y)(4 - 6y)}{(4x - 6y)^2}$$

Simplifying the expression, we get:

$$\frac{d^2y}{dx^2} = \frac{-8x + 12y + 8x + 24y - 24y^2}{(4x - 6y)^2}$$

Combining like terms, we get:

$$\frac{d^2y}{dx^2} = \frac{36y - 24y^2}{(4x - 6y)^2}$$

Conclusion

In this article, we used implicit differentiation to find the first and second derivatives of the given function $x^2 + 4xy - 3y^2 = 6$. The first derivative is $\frac{dy}{dx} = \frac{-2x - 4y}{4x - 6y}$, and the second derivative is $\frac{d^2y}{dx^2} = \frac{36y - 24y^2}{(4x - 6y)^2}$. Implicit differentiation is a powerful technique that can be used to find the derivatives of implicitly defined functions.

Example Use Cases

Implicit differentiation has numerous applications in various fields, including physics, engineering, and economics. Here are a few example use cases:

• Physics: Implicit differentiation can be used to find the velocity and acceleration of an object moving along a curved path.
• Engineering: Implicit differentiation can be used to find the stress and strain of a material under different loads.
• Economics: Implicit differentiation can be used to find the marginal cost and marginal revenue of a firm.

Conclusion

Implicit differentiation is a powerful technique used to find the derivatives of implicitly defined functions. In this article, we used implicit differentiation to find the first and second derivatives of the given function $x^2 + 4xy - 3y^2 = 6$. The first derivative is $\frac{dy}{dx} = \frac{-2x - 4y}{4x - 6y}$, and the second derivative is $\frac{d^2y}{dx^2} = \frac{36y - 24y^2}{(4x - 6y)^2}$. Implicit differentiation has numerous applications in various fields, including physics, engineering, and economics.

References

• Calculus: James Stewart, "Calculus: Early Transcendentals," 8th ed. (2015).
• Implicit Differentiation: Michael Corral, "Differential Equations for Engineers," 2nd ed. (2018).

Further Reading

• Implicit Differentiation: Khan Academy, "Implicit differentiation."
• Calculus: MIT OpenCourseWare, "Calculus."
• Differential Equations: Wolfram MathWorld, "Differential Equations."
  Implicit Differentiation: A Q&A Guide =====================================

Introduction

Implicit differentiation is a powerful technique used to find the derivatives of implicitly defined functions. In this article, we will answer some frequently asked questions about implicit differentiation.

Q: What is implicit differentiation?

A: 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.

Q: How do I apply implicit differentiation?

A: To apply implicit differentiation, you need to follow these steps:

1. Differentiate both sides of the equation with respect to the independent variable.
2. Treat the dependent variable as a function of the independent variable.
3. Use the chain rule to differentiate the terms involving the dependent variable.
4. Solve for the derivative of the dependent variable.

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

A: Here are some common mistakes to avoid when using implicit differentiation:

• Not treating the dependent variable as a function of the independent variable.
• Not using the chain rule to differentiate the terms involving the dependent variable.
• Not solving for the derivative of the dependent variable.
• Not checking the domain of the function.

Q: How do I find the second derivative using implicit differentiation?

A: To find the second derivative using implicit differentiation, you need to differentiate the first derivative with respect to the independent variable. This will give you the second derivative.

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

A: Implicit differentiation has numerous applications in various fields, including physics, engineering, and economics. Here are a few example use cases:

• Physics: Implicit differentiation can be used to find the velocity and acceleration of an object moving along a curved path.
• Engineering: Implicit differentiation can be used to find the stress and strain of a material under different loads.
• Economics: Implicit differentiation can be used to find the marginal cost and marginal revenue of a firm.

Q: How do I check the domain of the function using implicit differentiation?

A: To check the domain of the function using implicit differentiation, you need to ensure that the denominator of the derivative is not equal to zero. This will give you the domain of the function.

Q: What are some common functions that can be differentiated using implicit differentiation?

A: Here are some common functions that can be differentiated using implicit differentiation:

• Quadratic functions: $x^2 + 4xy - 3y^2 = 6$
• Cubic functions: $x^3 + 3x^2y - 2y^3 = 8$
• Polynomial functions: $x^4 + 2x^3y - 3y^4 = 10$

Conclusion

Implicit differentiation is a powerful technique used to find the derivatives of implicitly defined functions. In this article, we answered some frequently asked questions about implicit differentiation. We hope that this article has provided you with a better understanding of implicit differentiation and its applications.

References

• Calculus: James Stewart, "Calculus: Early Transcendentals," 8th ed. (2015).
• Implicit Differentiation: Michael Corral, "Differential Equations for Engineers," 2nd ed. (2018).

Further Reading

• Implicit Differentiation: Khan Academy, "Implicit differentiation."
• Calculus: MIT OpenCourseWare, "Calculus."
• Differential Equations: Wolfram MathWorld, "Differential Equations."