Differentiate.$ Y = \frac{e^x}{5-e^x} $Find $ Y^{\prime} $.

Feb 27, 2025 by ADMIN 60 views

$**Differentiate $y = \frac{e^x}{5-e^x}$: A Step-by-Step Guide**$

Introduction

In this article, we will delve into the world of calculus and explore the process of differentiating a complex function. Specifically, we will focus on finding the derivative of the function $y = \frac{e^x}{5-e^x}$ . This function is a rational function, and its derivative will involve the use of the quotient rule and the chain rule.

The Quotient Rule

Before we begin, let's recall the quotient rule, which is a fundamental rule in calculus for differentiating rational functions. The quotient rule states that if we have a function of the form $y = \frac{f(x)}{g(x)}$ , then the derivative of $y$ with respect to $x$ is given by:

y^{\prime} = \frac{f^{\prime}(x)g(x) - f(x)g^{\prime}(x)}{(g(x))^2}

Step 1: Identify the Functions $f(x)$ and $g(x)$

In our function $y = \frac{e^x}{5-e^x}$ , we can identify the functions $f(x)$ and $g(x)$ as follows:

f(x) = e^x

g(x) = 5 - e^x

Step 2: Find the Derivatives of $f(x)$ and $g(x)$

Next, we need to find the derivatives of $f(x)$ and $g(x)$ with respect to $x$ . Using the chain rule, we have:

f^{\prime}(x) = e^x

g^{\prime}(x) = -e^x

Step 3: Apply the Quotient Rule

Now that we have the derivatives of $f(x)$ and $g(x)$ , we can apply the quotient rule to find the derivative of $y$ with respect to $x$ :

y^{\prime} = \frac{f^{\prime}(x)g(x) - f(x)g^{\prime}(x)}{(g(x))^2}

y^{\prime} = \frac{(e^x)(5-e^x) - (e^x)(-e^x)}{(5-e^x)^2}

Simplify the Expression

To simplify the expression, we can expand the numerator and denominator:

y^{\prime} = \frac{5e^x - e^{2x} + e^{2x}}{(5-e^x)^2}

y^{\prime} = \frac{5e^x}{(5-e^x)^2}

Final Answer

Therefore, the derivative of the function $y = \frac{e^x}{5-e^x}$ is given by:

y^{\prime} = \frac{5e^x}{(5-e^x)^2}

Conclusion

In this article, we have shown how to differentiate the complex function $y = \frac{e^x}{5-e^x}$ using the quotient rule and the chain rule. The final answer is a rational function that involves the exponential function $e^x$ . This type of function is commonly encountered in calculus and is an important tool for modeling real-world phenomena.

Example Applications

The function $y = \frac{e^x}{5-e^x}$ has several example applications in mathematics and science. For instance, it can be used to model population growth or decay, where the exponential function $e^x$ represents the growth or decay rate. Additionally, it can be used to model chemical reactions, where the exponential function $e^x$ represents the concentration of a reactant or product.

References

Glossary

Quotient Rule: A rule in calculus for differentiating rational functions.
Chain Rule: A rule in calculus for differentiating composite functions.
Exponential Function: A function of the form $f(x) = e^x$ , where $e$ is a mathematical constant.

FAQs

Q: What is the derivative of the function $y = \frac{e^x}{5-e^x}$ ? A: The derivative of the function $y = \frac{e^x}{5-e^x}$ is given by $y^{\prime} = \frac{5e^x}{(5-e^x)^2}$ .
Q: What is the quotient rule? A: The quotient rule is a rule in calculus for differentiating rational functions.
Q: What is the chain rule? A: The chain rule is a rule in calculus for differentiating composite functions.
Differentiate $y = \frac{e^x}{5-e^x}$ : A Step-by-Step Guide ===========================================================

Q&A: Differentiate $y = \frac{e^x}{5-e^x}$

Q: What is the derivative of the function $y = \frac{e^x}{5-e^x}$ ?

A: The derivative of the function $y = \frac{e^x}{5-e^x}$ is given by $y^{\prime} = \frac{5e^x}{(5-e^x)^2}$ .

Q: What is the quotient rule?

A: The quotient rule is a rule in calculus for differentiating rational functions. It states that if we have a function of the form $y = \frac{f(x)}{g(x)}$ , then the derivative of $y$ with respect to $x$ is given by:

y^{\prime} = \frac{f^{\prime}(x)g(x) - f(x)g^{\prime}(x)}{(g(x))^2}

Q: What is the chain rule?

A: The chain rule is a rule in calculus for differentiating composite functions. It states that if we have a function of the form $y = f(g(x))$ , then the derivative of $y$ with respect to $x$ is given by:

y^{\prime} = f^{\prime}(g(x)) \cdot g^{\prime}(x)

Q: How do I apply the quotient rule to find the derivative of $y = \frac{e^x}{5-e^x}$ ?

A: To apply the quotient rule, we need to identify the functions $f(x)$ and $g(x)$ , and then find their derivatives. In this case, we have:

f(x) = e^x

g(x) = 5 - e^x

We then find the derivatives of $f(x)$ and $g(x)$ with respect to $x$ :

f^{\prime}(x) = e^x

g^{\prime}(x) = -e^x

We can then apply the quotient rule to find the derivative of $y$ with respect to $x$ :

y^{\prime} = \frac{f^{\prime}(x)g(x) - f(x)g^{\prime}(x)}{(g(x))^2}

y^{\prime} = \frac{(e^x)(5-e^x) - (e^x)(-e^x)}{(5-e^x)^2}

Q: How do I simplify the expression for the derivative of $y = \frac{e^x}{5-e^x}$ ?

A: To simplify the expression, we can expand the numerator and denominator:

y^{\prime} = \frac{5e^x - e^{2x} + e^{2x}}{(5-e^x)^2}

y^{\prime} = \frac{5e^x}{(5-e^x)^2}

Q: What are some example applications of the function $y = \frac{e^x}{5-e^x}$ ?

A: The function $y = \frac{e^x}{5-e^x}$ has several example applications in mathematics and science. For instance, it can be used to model population growth or decay, where the exponential function $e^x$ represents the growth or decay rate. Additionally, it can be used to model chemical reactions, where the exponential function $e^x$ represents the concentration of a reactant or product.

Q: Where can I find more information on the quotient rule and the chain rule?

A: For further reading on the topic of differentiation and the quotient rule, we recommend the following resources:

Q: What is the exponential function?

A: The exponential function is a function of the form $f(x) = e^x$ , where $e$ is a mathematical constant. The exponential function is used to model growth or decay, and is a fundamental concept in mathematics and science.

Q: What is the difference between the quotient rule and the chain rule?

A: The quotient rule is used to differentiate rational functions, while the chain rule is used to differentiate composite functions. The quotient rule involves the use of the derivative of the numerator and denominator, while the chain rule involves the use of the derivative of the outer function and the inner function.

Q: How do I use the quotient rule to find the derivative of a rational function?

A: To use the quotient rule, we need to identify the functions $f(x)$ and $g(x)$ , and then find their derivatives. We can then apply the quotient rule to find the derivative of the rational function.

Q: How do I use the chain rule to find the derivative of a composite function?

A: To use the chain rule, we need to identify the outer function and the inner function, and then find their derivatives. We can then apply the chain rule to find the derivative of the composite function.

Q: What are some common mistakes to avoid when using the quotient rule and the chain rule?

A: Some common mistakes to avoid when using the quotient rule and the chain rule include:

Failing to identify the functions $f(x)$ and $g(x)$ correctly
Failing to find the derivatives of $f(x)$ and $g(x)$ correctly
Failing to apply the quotient rule or the chain rule correctly
Failing to simplify the expression for the derivative correctly

Q: How do I check my work when using the quotient rule and the chain rule?

A: To check your work, we recommend the following steps:

Verify that you have identified the functions $f(x)$ and $g(x)$ correctly
Verify that you have found the derivatives of $f(x)$ and $g(x)$ correctly
Verify that you have applied the quotient rule or the chain rule correctly
Verify that you have simplified the expression for the derivative correctly

Q: What are some resources for further reading on the topic of differentiation and the quotient rule?