Find The Slope Of The Tangent Line To The Curve$\[ 4x^2 + 2xy - 4y^3 = 52 \\]at The Point \[$(-4,1)\$\].

Introduction

Implicit differentiation is a powerful technique used to find the derivative of an implicitly defined function. In this article, we will use implicit differentiation to find the slope of the tangent line to a given curve at a specific point. The curve is defined by the equation $4x^2 + 2xy - 4y^3 = 52$, and we want to find the slope of the tangent line at the point $(-4,1)$.

Implicit Differentiation

Implicit differentiation is a method of differentiating 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. In this case, we have the equation $4x^2 + 2xy - 4y^3 = 52$, and we want to find the derivative of $y$ with respect to $x$.

To do this, 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 $y$.

Differentiating the Equation

Let's start by differentiating the equation $4x^2 + 2xy - 4y^3 = 52$ with respect to $x$. We will use the following rules:

• The derivative of $x^n$ is $nx^{n-1}$.
• The derivative of $xy$ is $y + x\frac{dy}{dx}$.
• The derivative of $y^3$ is $3y^2\frac{dy}{dx}$.

Applying these rules, we get:

$$\frac{d}{dx}(4x^2) + \frac{d}{dx}(2xy) - \frac{d}{dx}(4y^3) = \frac{d}{dx}(52)$$

$$8x + 2y + 2x\frac{dy}{dx} - 12y^2\frac{dy}{dx} = 0$$

Simplifying the Equation

Now, we can simplify the equation by combining like terms:

$$8x + 2y + 2x\frac{dy}{dx} - 12y^2\frac{dy}{dx} = 0$$

$$\frac{dy}{dx}(2x - 12y^2) = -8x - 2y$$

Finding the Slope of the Tangent Line

Now, we can find the slope of the tangent line by dividing both sides of the equation by $2x - 12y^2$:

$$\frac{dy}{dx} = \frac{-8x - 2y}{2x - 12y^2}$$

Evaluating the Slope at the Given Point

We want to find the slope of the tangent line at the point $(-4,1)$. To do this, we will substitute $x = -4$ and $y = 1$ into the equation:

$$\frac{dy}{dx} = \frac{-8(-4) - 2(1)}{2(-4) - 12(1)^2}$$

$$\frac{dy}{dx} = \frac{32 - 2}{-8 - 12}$$

$$\frac{dy}{dx} = \frac{30}{-20}$$

$$\frac{dy}{dx} = -\frac{3}{2}$$

Conclusion

In this article, we used implicit differentiation to find the slope of the tangent line to a given curve at a specific point. The curve was defined by the equation $4x^2 + 2xy - 4y^3 = 52$, and we wanted to find the slope of the tangent line at the point $(-4,1)$. We differentiated both sides of the equation with respect to $x$, used the product rule and the chain rule to differentiate the terms involving $y$, and finally found the slope of the tangent line by dividing both sides of the equation by $2x - 12y^2$. The slope of the tangent line at the point $(-4,1)$ is $-\frac{3}{2}$.

Implicit Differentiation: A Powerful Tool

Implicit differentiation is a powerful tool 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. In this article, we used implicit differentiation to find the slope of the tangent line to a given curve at a specific point. The technique is widely used in mathematics and physics to solve problems involving implicitly defined functions.

Applications of Implicit Differentiation

Implicit differentiation has many applications in mathematics and physics. Some of the applications include:

• Finding the derivative of an implicitly defined function.
• Finding the slope of the tangent line to a curve at a specific point.
• Solving problems involving implicitly defined functions.
• Finding the equation of a tangent line to a curve at a specific point.

Conclusion

Introduction

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

Q: What is implicit differentiation?

A: Implicit differentiation is a method of differentiating 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: When is implicit differentiation used?

A: Implicit differentiation is used when the function is defined implicitly, meaning that it is not possible to isolate the dependent variable. This is often the case in problems involving curves, surfaces, and other geometric shapes.

Q: How do I apply implicit differentiation?

A: To apply 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. Simplify the resulting equation to find 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:

• Failing to differentiate both sides of the equation.
• Failing to use the product rule and the chain rule when differentiating terms involving the dependent variable.
• Simplifying the equation incorrectly.

Q: Can implicit differentiation be used to find the equation of a tangent line?

A: Yes, implicit differentiation can be used to find the equation of a tangent line. To do this, find the derivative of the dependent variable and then use the point-slope form of a line to write the equation of the tangent line.

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

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

• Finding the derivative of an implicitly defined function.
• Finding the slope of the tangent line to a curve at a specific point.
• Solving problems involving implicitly defined functions.
• Finding the equation of a tangent line to a curve at a specific point.

Q: How do I know if a function is implicitly defined?

A: A function is implicitly defined if it is not possible to isolate the dependent variable. This is often the case in problems involving curves, surfaces, and other geometric shapes.

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

A: No, implicit differentiation can only be used to find the derivative of a function that is implicitly defined.

Q: What are some common types of problems that involve implicit differentiation?

A: Some common types of problems that involve implicit differentiation include:

• Finding the derivative of an implicitly defined function.
• Finding the slope of the tangent line to a curve at a specific point.
• Solving problems involving implicitly defined functions.
• Finding the equation of a tangent line to a curve at a specific point.

Conclusion

Implicit differentiation is a powerful technique used to find the derivative of an implicitly defined function. In this article, we have answered some of the most frequently asked questions about implicit differentiation. We hope that this guide has been helpful in understanding the concept of implicit differentiation and how to apply it to solve problems.