Differentiate.$\[ \begin{array}{c} y=\frac{e^x}{2-e^x} \\ y^{\prime}=\square \end{array} \\]

Feb 28, 2025 by ADMIN 93 views

Differentiate: A Comprehensive Guide to Solving the Given Equation

Introduction

In mathematics, differentiation is a fundamental concept that plays a crucial role in various fields, including calculus, physics, and engineering. It involves finding the derivative of a function, which represents the rate of change of the function with respect to its input. In this article, we will focus on differentiating the given equation, which is a rational function involving exponential terms.

The Given Equation

The given equation is:

y = \frac{e^x}{2-ex}

This equation represents a rational function, where the numerator is an exponential function and the denominator is a polynomial function. To differentiate this equation, we will use the quotient rule of differentiation, which states that if y = \frac{f(x)}{g(x)}, then y' = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}.

Differentiating the Equation

To differentiate the given equation, we will first find the derivatives of the numerator and denominator separately.

Derivative of the Numerator

The numerator is an exponential function, e^x. The derivative of e^x is also e^x, since the derivative of e^x is e^x.

Derivative of the Denominator

The denominator is a polynomial function, 2-e^x. To find its derivative, we will use the chain rule of differentiation, which states that if y = f(g(x)), then y' = f'(g(x)) * g'(x). In this case, f(u) = 2-u and g(x) = e^x. Therefore, f'(u) = -1 and g'(x) = e^x. The derivative of the denominator is:

\frac{d}{dx}(2-e^x) = -e^x

Applying the Quotient Rule

Now that we have found the derivatives of the numerator and denominator, we can apply the quotient rule to find the derivative of the given equation.

y' = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}

Substituting the values of f(x), g(x), f'(x), and g'(x), we get:

y' = \frac{e^x(2-ex) - e^x(-ex)}{(2-e^x)2}

Simplifying the expression, we get:

y' = \frac{e^x(2-ex + e^x)}{(2-ex)^2}

y' = \frac{e^x(2)}{(2-ex)^2}

y' = \frac{2e^x}{(2-ex)^2}

Conclusion

In this article, we have differentiated the given equation, which is a rational function involving exponential terms. We have used the quotient rule of differentiation to find the derivative of the equation. The final derivative is:

y' = \frac{2e^x}{(2-ex)^2}

This derivative represents the rate of change of the given equation with respect to its input. It can be used to analyze the behavior of the equation and make predictions about its future values.

Applications of Differentiation

Differentiation has numerous applications in various fields, including:

Physics: Differentiation is used to describe the motion of objects, including their velocity and acceleration.
Engineering: Differentiation is used to design and optimize systems, including electrical circuits and mechanical systems.
Economics: Differentiation is used to analyze the behavior of economic systems, including the supply and demand of goods and services.
Biology: Differentiation is used to study the growth and development of living organisms, including the rate of cell division and the rate of protein synthesis.

Real-World Examples of Differentiation

Differentiation has numerous real-world applications, including:

Predicting Stock Prices: Differentiation can be used to predict the future values of stock prices based on their past behavior.
Designing Electrical Circuits: Differentiation can be used to design and optimize electrical circuits, including the placement of resistors and capacitors.
Analyzing the Behavior of Economic Systems: Differentiation can be used to analyze the behavior of economic systems, including the supply and demand of goods and services.
Studying the Growth and Development of Living Organisms: Differentiation can be used to study the growth and development of living organisms, including the rate of cell division and the rate of protein synthesis.

Conclusion

In conclusion, differentiation is a fundamental concept in mathematics that plays a crucial role in various fields, including calculus, physics, and engineering. It involves finding the derivative of a function, which represents the rate of change of the function with respect to its input. In this article, we have differentiated the given equation, which is a rational function involving exponential terms. The final derivative is:

y' = \frac{2e^x}{(2-ex)^2}

This derivative represents the rate of change of the given equation with respect to its input. It can be used to analyze the behavior of the equation and make predictions about its future values.

Introduction

In our previous article, we discussed the differentiation of the given equation, which is a rational function involving exponential terms. We used the quotient rule of differentiation to find the derivative of the equation. In this article, we will answer some frequently asked questions related to the differentiation of the given equation.

Q&A

Q: What is the quotient rule of differentiation?

A: The quotient rule of differentiation is a rule that is used to find the derivative of a rational function. It states that if y = \frac{f(x)}{g(x)}, then y' = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}.

Q: How do I apply the quotient rule to find the derivative of the given equation?

A: To apply the quotient rule, you need to find the derivatives of the numerator and denominator separately. Then, you can use the quotient rule formula to find the derivative of the given equation.

Q: What is the derivative of the numerator of the given equation?

A: The derivative of the numerator is e^x, since the derivative of e^x is e^x.

Q: What is the derivative of the denominator of the given equation?

A: The derivative of the denominator is -e^x, since the derivative of 2-e^x is -e^x.

Q: How do I simplify the expression for the derivative of the given equation?

A: To simplify the expression, you can cancel out any common factors in the numerator and denominator.

Q: What is the final derivative of the given equation?

A: The final derivative of the given equation is:

y' = \frac{2e^x}{(2-ex)^2}

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

A: Differentiation has numerous real-world applications, including predicting stock prices, designing electrical circuits, analyzing the behavior of economic systems, and studying the growth and development of living organisms.

Q: How do I use differentiation to analyze the behavior of economic systems?

A: To use differentiation to analyze the behavior of economic systems, you can use the derivative of a function to describe the rate of change of the function with respect to its input. This can help you understand how the system will behave in the future.

Q: How do I use differentiation to study the growth and development of living organisms?

A: To use differentiation to study the growth and development of living organisms, you can use the derivative of a function to describe the rate of change of the function with respect to its input. This can help you understand how the organism will grow and develop over time.

Conclusion

In conclusion, differentiation is a fundamental concept in mathematics that plays a crucial role in various fields, including calculus, physics, and engineering. It involves finding the derivative of a function, which represents the rate of change of the function with respect to its input. In this article, we have answered some frequently asked questions related to the differentiation of the given equation. We hope that this article has been helpful in understanding the concept of differentiation and its applications.

Additional Resources

If you are interested in learning more about differentiation, we recommend the following resources:

Calculus Textbooks: There are many excellent calculus textbooks available that cover the topic of differentiation in detail.
Online Resources: There are many online resources available that provide tutorials and examples on differentiation, including Khan Academy and MIT OpenCourseWare.
Mathematical Software: There are many mathematical software packages available that can be used to visualize and analyze the behavior of functions, including Mathematica and MATLAB.

Final Thoughts

In conclusion, differentiation is a powerful tool that can be used to analyze and understand the behavior of functions. It has numerous real-world applications, including predicting stock prices, designing electrical circuits, analyzing the behavior of economic systems, and studying the growth and development of living organisms. We hope that this article has been helpful in understanding the concept of differentiation and its applications.