Differentiate The Function.${ Y = \frac{1}{(4x - 9)^8} }$ {\frac{dy}{dx} = \, \boxed{\text{(Simplify Your Answer.)}}\}

Introduction

In this article, we will delve into the world of calculus and explore the process of differentiating a given function. The function in question is $y = \frac{1}{(4x - 9)^8}$. Our goal is to find the derivative of this function, denoted as $\frac{dy}{dx}$.

Understanding the Function

Before we proceed with the differentiation process, let's take a closer look at the given function. The function is a rational function, which means it is the ratio of two polynomials. In this case, the numerator is a constant (1), and the denominator is a polynomial raised to the power of 8.

$y = \frac{1}{(4x - 9)^8}$

The denominator of the function is a polynomial of degree 1, which means it is a linear function. The polynomial is $4x - 9$, and it is raised to the power of 8.

Applying the Chain Rule

To differentiate the function, we will apply the chain rule. The chain rule is a fundamental rule in calculus that allows us to differentiate composite functions. In this case, the function is a composite function, where the outer function is the reciprocal function, and the inner function is the polynomial $4x - 9$.

To apply the chain rule, we need to find the derivative of the outer function and the derivative of the inner function. The derivative of the outer function is the derivative of the reciprocal function, which is $-\frac{1}{u^2}$, where $u$ is the inner function.

The derivative of the inner function is the derivative of the polynomial $4x - 9$, which is $4$.

Finding the Derivative

Now that we have the derivatives of the outer and inner functions, we can apply the chain rule to find the derivative of the function.

$\frac{dy}{dx} = -\frac{1}{u^2} \cdot \frac{du}{dx}$

Substituting the values of $u$ and $\frac{du}{dx}$, we get:

$\frac{dy}{dx} = -\frac{1}{(4x - 9)^8} \cdot 4$

Simplifying the expression, we get:

$\frac{dy}{dx} = -\frac{4}{(4x - 9)^8}$

Conclusion

In this article, we differentiated the function $y = \frac{1}{(4x - 9)^8}$ using the chain rule. We found the derivative of the function to be $\frac{dy}{dx} = -\frac{4}{(4x - 9)^8}$. This derivative represents the rate of change of the function with respect to the variable $x$.

Step-by-Step Solution

Here is the step-by-step solution to the problem:

1. Apply the chain rule: The function is a composite function, where the outer function is the reciprocal function, and the inner function is the polynomial $4x - 9$.
2. Find the derivative of the outer function: The derivative of the outer function is the derivative of the reciprocal function, which is $-\frac{1}{u^2}$, where $u$ is the inner function.
3. Find the derivative of the inner function: The derivative of the inner function is the derivative of the polynomial $4x - 9$, which is $4$.
4. Apply the chain rule: Substitute the values of $u$ and $\frac{du}{dx}$ into the chain rule formula.
5. Simplify the expression: Simplify the resulting expression to get the final derivative.

Final Answer

The final answer is:

$$

Introduction

In our previous article, we differentiated the function $y = \frac{1}{(4x - 9)^8}$ using the chain rule. In this article, we will provide a Q&A guide to help you understand the process of differentiating the function.

Q: What is the chain rule?

A: The chain rule is a fundamental rule in calculus that allows us to differentiate composite functions. A composite function is a function that is composed of two or more functions. In this case, the function is a composite function, where the outer function is the reciprocal function, and the inner function is the polynomial $4x - 9$.

Q: How do I apply the chain rule?

A: To apply the chain rule, you need to find the derivative of the outer function and the derivative of the inner function. The derivative of the outer function is the derivative of the reciprocal function, which is $-\frac{1}{u^2}$, where $u$ is the inner function. The derivative of the inner function is the derivative of the polynomial $4x - 9$, which is $4$.

Q: What is the derivative of the reciprocal function?

A: The derivative of the reciprocal function is $-\frac{1}{u^2}$, where $u$ is the inner function.

Q: What is the derivative of the polynomial $4x - 9$?

A: The derivative of the polynomial $4x - 9$ is $4$.

Q: How do I simplify the expression?

A: To simplify the expression, you need to substitute the values of $u$ and $\frac{du}{dx}$ into the chain rule formula. Then, you need to simplify the resulting expression to get the final derivative.

Q: What is the final derivative of the function?

A: The final derivative of the function is $\frac{dy}{dx} = -\frac{4}{(4x - 9)^8}$.

Q: Why is the derivative of the function important?

A: The derivative of the function is important because it represents the rate of change of the function with respect to the variable $x$. This is useful in many applications, such as physics, engineering, and economics.

Q: How do I use the derivative of the function in real-world applications?

A: The derivative of the function can be used in many real-world applications, such as:

• Physics: The derivative of the function can be used to describe the motion of an object.
• Engineering: The derivative of the function can be used to design and optimize systems.
• Economics: The derivative of the function can be used to model and analyze economic systems.

Conclusion

In this article, we provided a Q&A guide to help you understand the process of differentiating the function $y = \frac{1}{(4x - 9)^8}$. We hope that this guide has been helpful in clarifying any questions you may have had about the process.

Step-by-Step Solution

Here is the step-by-step solution to the problem:

1. Apply the chain rule: The function is a composite function, where the outer function is the reciprocal function, and the inner function is the polynomial $4x - 9$.
2. Find the derivative of the outer function: The derivative of the outer function is the derivative of the reciprocal function, which is $-\frac{1}{u^2}$, where $u$ is the inner function.
3. Find the derivative of the inner function: The derivative of the inner function is the derivative of the polynomial $4x - 9$, which is $4$.
4. Apply the chain rule: Substitute the values of $u$ and $\frac{du}{dx}$ into the chain rule formula.
5. Simplify the expression: Simplify the resulting expression to get the final derivative.

Final Answer

The final answer is:

$\boxed{-\frac{4}{(4x - 9)^8}}$