Use Implicit Differentiation Of The Equation Below To Determine The Slope Of The Graph At The Given Point.$\[ X Y^3 = 16 \\]Given Point: $\[ X = -\frac{1}{4}, Y = -4 \\]The Slope Of The Graph At The Given Point Is

Introduction

Implicit differentiation is a powerful tool in calculus that allows us to find the derivative of an implicitly defined function. In this article, we will use implicit differentiation to find the slope of the graph of the equation $x y^3 = 16$ at the given point $x = -\frac{1}{4}, y = -4$.

Implicit Differentiation

Implicit differentiation is a technique 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.

To find the derivative of the equation $x y^3 = 16$, we will differentiate both sides of the equation with respect to $x$. We will use the chain rule to differentiate the term $y^3$.

Step 1: Differentiate Both Sides of the Equation

We will start by differentiating both sides of the equation $x y^3 = 16$ with respect to $x$. We will use the product rule to differentiate the term $x y^3$.

$$\frac{d}{dx} (x y^3) = \frac{d}{dx} (16)$$

Using the product rule, we get:

$$\frac{d}{dx} (x y^3) = y^3 + x \frac{d}{dx} (y^3)$$

Now, we will differentiate the term $y^3$ with respect to $x$. We will use the chain rule to do this.

$$\frac{d}{dx} (y^3) = 3 y^2 \frac{dy}{dx}$$

Substituting this into the previous equation, we get:

$$y^3 + x 3 y^2 \frac{dy}{dx} = 0$$

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

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

$$x 3 y^2 \frac{dy}{dx} = -y^3$$

Dividing both sides of the equation by $x 3 y^2$, we get:

$$\frac{dy}{dx} = -\frac{y^3}{x 3 y^2}$$

Simplifying the equation, we get:

$$\frac{dy}{dx} = -\frac{y}{3x}$$

Finding the Slope at the Given Point

We are given the point $x = -\frac{1}{4}, y = -4$. We will substitute these values into the equation for $\frac{dy}{dx}$ to find the slope of the graph at this point.

$$\frac{dy}{dx} = -\frac{(-4)}{3(-\frac{1}{4})}$$

Simplifying the equation, we get:

$$\frac{dy}{dx} = -\frac{(-4)}{(-\frac{3}{4})}$$

$$\frac{dy}{dx} = \frac{16}{3}$$

Therefore, the slope of the graph at the given point is $\frac{16}{3}$.

Conclusion

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Introduction

Implicit differentiation is a powerful tool in calculus that allows us to find the derivative of an implicitly defined function. In this article, we will answer some common questions about implicit differentiation.

Q: What is implicit differentiation?

A: Implicit differentiation is a technique 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 use implicit differentiation to find the derivative of an implicitly defined function?

A: To use implicit differentiation, you will 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. Solve for the derivative of the dependent variable.

Q: What is the product rule in implicit differentiation?

A: The product rule in implicit differentiation is used to differentiate the product of two functions. It states that if we have a function of the form $f(x)g(x)$, then the derivative of this function is given by:

$$\frac{d}{dx} (f(x)g(x)) = f(x)\frac{dg}{dx} + g(x)\frac{df}{dx}$$

Q: What is the chain rule in implicit differentiation?

A: The chain rule in implicit differentiation is used to differentiate composite functions. It states that if we have a function of the form $f(g(x))$, then the derivative of this function is given by:

$$\frac{d}{dx} (f(g(x))) = f'(g(x)) \cdot g'(x)$$

Q: How do I use implicit differentiation to find the slope of a graph at a given point?

A: To use implicit differentiation to find the slope of a graph at a given point, you will 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. Solve for the derivative of the dependent variable.
4. Substitute the given values into the equation for the derivative of the dependent variable.
5. Simplify the equation to find the slope of the graph at the given point.

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 treat the dependent variable as a function of the independent variable.
• Forgetting to use the product rule and chain rule when differentiating composite functions.
• Not simplifying the equation after finding the derivative of the dependent variable.
• Not checking the domain of the function before finding the derivative.

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

A: Implicit differentiation has many real-world applications, including:

• Finding the slope of a graph at a given point.
• Finding the equation of a tangent line to a curve.
• Finding the equation of a normal line to a curve.
• Finding the maximum or minimum value of a function.

Conclusion

Implicit differentiation is a powerful tool in calculus that allows us to find the derivative of an implicitly defined function. In this article, we answered some common questions about implicit differentiation, including how to use it to find the derivative of an implicitly defined function, how to use it to find the slope of a graph at a given point, and some common mistakes to avoid when using it. We also discussed some real-world applications of implicit differentiation.