Use Implicit Differentiation Of The Equation Below To Determine The Slope Of The Graph At The Given Point.$\[ X Y^3 = 2 \\]Given: \[$ X = -\frac{1}{4}, Y = -2 \$\]The Slope Of The Graph At The Given Point Is \[$\square\$\].

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 determine the slope of the graph of the equation $x y^3 = 2$ at the given point $x = -\frac{1}{4}, y = -2$.

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.

To differentiate the equation $x y^3 = 2$ with respect to $x$, we will use the product rule and the chain rule.

Step 1: Differentiate both sides of the equation

We will start by differentiating both sides of the equation with respect to $x$. This will give us:

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

Using the product rule, we can rewrite the left-hand side of the equation as:

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

Step 2: Apply the chain rule

To differentiate $y^3$, we will use the chain rule. The chain rule states that if we have a composite function of the form $f(g(x))$, then the derivative of this function is given by:

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

In this case, we have $f(y) = y^3$ and $g(x) = x$. Therefore, we can rewrite the derivative of $y^3$ as:

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

Step 3: Substitute the derivative of y

We know that $y$ is a function of $x$, so we can substitute the derivative of $y$ into the equation. Let's call the derivative of $y$ with respect to $x$ as $\frac{dy}{dx}$. Then, we can rewrite the equation as:

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

Step 4: Solve for dy/dx

Now, we can solve for $\frac{dy}{dx}$. To do this, we will isolate $\frac{dy}{dx}$ on one side of the equation. We can do this by subtracting $y^3$ from both sides of the equation and then dividing both sides by $3 x y^2$.

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

Step 5: Simplify the expression

We can simplify the expression for $\frac{dy}{dx}$ by canceling out the common factors. We can cancel out the $y^2$ term in the numerator and denominator.

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

Finding the Slope at the Given Point

Now that we have the expression for $\frac{dy}{dx}$, we can use it to find the slope of the graph at the given point $x = -\frac{1}{4}, y = -2$.

We will substitute the values of $x$ and $y$ into the expression for $\frac{dy}{dx}$.

$$\frac{dy}{dx} = -\frac{-2}{3 \left(-\frac{1}{4}\right)}$$

Simplifying the expression, we get:

$$\frac{dy}{dx} = -\frac{-2}{-\frac{3}{4}}$$

$$\frac{dy}{dx} = \frac{8}{3}$$

Therefore, the slope of the graph at the given point is $\boxed{\frac{8}{3}}$.

Conclusion

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Introduction

Implicit differentiation is a powerful technique used to find the derivative of an implicitly defined function. In this article, we will answer some common questions about implicit differentiation and provide examples to illustrate the concept.

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: 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. Apply the product rule and the chain rule as needed.
3. Solve for the derivative of the dependent variable.

Q: What is the product rule?

A: The product rule is a rule for differentiating a 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:

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

Q: What is the chain rule?

A: The chain rule is a rule for differentiating a composite function. It states that if we have a composite 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 find the derivative of a function using implicit differentiation?

A: To find the derivative of a function using implicit differentiation, you need to follow these steps:

1. Differentiate both sides of the equation with respect to the independent variable.
2. Apply the product rule and the chain rule as needed.
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 apply the product rule and the chain rule.
• Not solving for the derivative of the dependent variable.
• Making algebraic errors when simplifying the expression.

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

A: No, implicit differentiation is only used to find the derivative of an implicitly defined function. If the function is not implicitly defined, you need to use a different method to find its 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 curve at a given point.
• Determining the rate of change of a quantity.
• Modeling physical systems, such as the motion of an object.

Conclusion

Implicit differentiation is a powerful technique used to find the derivative of an implicitly defined function. By following the steps outlined in this article, you can use implicit differentiation to find the derivative of a function and solve a variety of problems. Remember to avoid common mistakes and to apply the product rule and the chain rule as needed.