Suppose That X And Y Are Related By The Given Equation. Use Implicit Differentiation To Determine $\frac{dy}{dx}$.$y^6 - 5x^5 = 7x$\frac{dy}{dx} = \square$

Introduction

Implicit differentiation is a technique used to find the derivative of an implicitly defined function. In this case, we are given the equation $y^6 - 5x^5 = 7x$ and we need to use implicit differentiation to determine $\frac{dy}{dx}$. This technique is essential in calculus and is used to solve a wide range of problems in mathematics and physics.

The Process of Implicit Differentiation

Implicit differentiation involves differentiating both sides of the equation with respect to $x$, while treating $y$ as a function of $x$. This means that we will use the chain rule to differentiate the terms involving $y$. The process of implicit differentiation can be summarized as follows:

1. Differentiate both sides of the equation with respect to $x$.
2. Use the chain rule to differentiate the terms involving $y$.
3. Simplify the resulting equation to obtain the derivative of $y$ with respect to $x$.

Applying Implicit Differentiation to the Given Equation

Now, let's apply the process of implicit differentiation to the given equation $y^6 - 5x^5 = 7x$. We will start by differentiating both sides of the equation with respect to $x$.

Step 1: Differentiate Both Sides of the Equation

To differentiate both sides of the equation, we will use the power rule of differentiation, which states that if $f(x) = x^n$, then $f'(x) = nx^{n-1}$. We will also use the chain rule to differentiate the terms involving $y$.

\frac{d}{dx} (y^6) = 6y^5 \frac{dy}{dx}
\frac{d}{dx} (-5x^5) = -25x^4
\frac{d}{dx} (7x) = 7

Step 2: Simplify the Resulting Equation

Now, we will simplify the resulting equation by combining like terms.

6y^5 \frac{dy}{dx} - 25x^4 = 7

Step 3: Solve for $\frac{dy}{dx}$

To solve for $\frac{dy}{dx}$, we will isolate the term involving $\frac{dy}{dx}$ on one side of the equation.

6y^5 \frac{dy}{dx} = 25x^4 + 7
\frac{dy}{dx} = \frac{25x^4 + 7}{6y^5}

Conclusion

In this article, we used implicit differentiation to determine $\frac{dy}{dx}$ for the given equation $y^6 - 5x^5 = 7x$. We started by differentiating both sides of the equation with respect to $x$, then used the chain rule to differentiate the terms involving $y$. Finally, we simplified the resulting equation to obtain the derivative of $y$ with respect to $x$. The final answer is $\frac{dy}{dx} = \frac{25x^4 + 7}{6y^5}$.

Example Use Cases

Implicit differentiation has many practical applications in mathematics and physics. Here are a few example use cases:

• Physics: Implicit differentiation can be used to solve problems involving motion, such as finding the velocity and acceleration of an object.
• Engineering: Implicit differentiation can be used to solve problems involving optimization, such as finding the maximum or minimum of a function.
• Economics: Implicit differentiation can be used to solve problems involving economics, such as finding the demand and supply curves of a product.

Conclusion

Q&A: 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 $x$, while treating $y$ as a function of $x$.

Q: When should I use implicit differentiation?

A: You should use implicit differentiation when the equation is not easily solvable for $y$ in terms of $x$. This is often the case when the equation involves a function of $y$ that is not easily invertible.

Q: How do I apply implicit differentiation to a given equation?

A: To apply implicit differentiation to a given equation, follow these steps:

1. Differentiate both sides of the equation with respect to $x$.
2. Use the chain rule to differentiate the terms involving $y$.
3. Simplify the resulting equation to obtain the derivative of $y$ with respect to $x$.

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 use the chain rule when differentiating terms involving $y$.
• Not simplifying the resulting equation to obtain the derivative of $y$ with respect to $x$.
• Not checking the domain of the function to ensure that the derivative is defined.

Q: Can I use implicit differentiation to find the second derivative of a function?

A: Yes, you can use implicit differentiation to find the second derivative of a function. To do this, you will need to differentiate the first derivative with respect to $x$.

Q: How do I use implicit differentiation to solve optimization problems?

A: To use implicit differentiation to solve optimization problems, follow these steps:

1. Define the function that you want to optimize.
2. Use implicit differentiation to find the derivative of the function with respect to $x$.
3. Set the derivative equal to zero and solve for $x$.
4. Check the second derivative to ensure that the function has a maximum or minimum at the critical point.

Q: Can I use implicit differentiation to solve problems involving motion?

A: Yes, you can use implicit differentiation to solve problems involving motion. To do this, you will need to use the chain rule to differentiate the terms involving $y$.

Q: How do I use implicit differentiation to solve problems involving economics?

A: To use implicit differentiation to solve problems involving economics, follow these steps:

1. Define the function that you want to analyze.
2. Use implicit differentiation to find the derivative of the function with respect to $x$.
3. Set the derivative equal to zero and solve for $x$.
4. Check the second derivative to ensure that the function has a maximum or minimum at the critical point.

Conclusion

Implicit differentiation is a powerful tool for finding derivatives of implicitly defined functions. By following the steps outlined in this article, you can use implicit differentiation to solve a wide range of problems in mathematics and physics. Remember to always start by differentiating both sides of the equation with respect to $x$, then use the chain rule to differentiate the terms involving $y$. Finally, simplify the resulting equation to obtain the derivative of $y$ with respect to $x$.