Find The Derivative Of The Following Expression With Respect To \[$x\$\]:$\[ \frac{d}{dx}\left(\frac{1-x E^x}{x+e^x}\right) \\]
Introduction
In calculus, finding the derivative of a function is a crucial concept that helps us understand how the function changes as its input changes. In this article, we will focus on finding the derivative of a complex expression with respect to a variable x. We will use the quotient rule and chain rule of differentiation to simplify the expression and find its derivative.
The Quotient Rule
The quotient rule is a fundamental rule in calculus that helps us find the derivative of a quotient of two functions. If we have a function of the form:
f(x) = g(x) / h(x)
where g(x) and h(x) are both functions of x, then the derivative of f(x) with respect to x is given by:
f'(x) = (h(x)g'(x) - g(x)h'(x)) / h(x)^2
The Chain Rule
The chain rule is another important rule in calculus that helps us find the derivative of a composite function. If we have a function of the form:
f(x) = g(h(x))
where g(x) and h(x) are both functions of x, then the derivative of f(x) with respect to x is given by:
f'(x) = g'(h(x)) * h'(x)
Finding the Derivative of the Given Expression
Now that we have covered the quotient rule and chain rule, let's apply these rules to find the derivative of the given expression:
d/dx (1 - xe^x) / (x + e^x)
To find the derivative of this expression, we will use the quotient rule. We can rewrite the expression as:
f(x) = (1 - xe^x) / (x + e^x)
where g(x) = 1 - xe^x and h(x) = x + e^x.
Step 1: Find the Derivative of the Numerator
To find the derivative of the numerator, we will use the product rule and chain rule. The product rule states that if we have a function of the form:
f(x) = u(x)v(x)
where u(x) and v(x) are both functions of x, then the derivative of f(x) with respect to x is given by:
f'(x) = u'(x)v(x) + u(x)v'(x)
In this case, we have:
u(x) = 1 v(x) = -xe^x
The derivative of u(x) with respect to x is:
u'(x) = 0
The derivative of v(x) with respect to x is:
v'(x) = -e^x - xe^x
Using the product rule, we get:
f'(x) = u'(x)v(x) + u(x)v'(x) = 0 * (-xe^x) + 1 * (-e^x - xe^x) = -e^x - xe^x
Step 2: Find the Derivative of the Denominator
To find the derivative of the denominator, we will use the sum rule and chain rule. The sum rule states that if we have a function of the form:
f(x) = u(x) + v(x)
where u(x) and v(x) are both functions of x, then the derivative of f(x) with respect to x is given by:
f'(x) = u'(x) + v'(x)
In this case, we have:
u(x) = x v(x) = e^x
The derivative of u(x) with respect to x is:
u'(x) = 1
The derivative of v(x) with respect to x is:
v'(x) = e^x
Using the sum rule, we get:
f'(x) = u'(x) + v'(x) = 1 + e^x
Step 3: Apply 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 expression:
f'(x) = (h(x)g'(x) - g(x)h'(x)) / h(x)^2
where g(x) = 1 - xe^x and h(x) = x + e^x.
Substituting the values of g(x), h(x), g'(x), and h'(x), we get:
f'(x) = ((x + ex)(-ex - xe^x) - (1 - xe^x)(1 + e^x)) / (x + ex)2
Simplifying the expression, we get:
f'(x) = (-xe^x - e^x - x2ex - xe^x) / (x + ex)2
Combining like terms, we get:
f'(x) = (-x2ex - 2xe^x - e^x) / (x + ex)2
Conclusion
In this article, we have found the derivative of a complex expression with respect to a variable x. We used the quotient rule and chain rule of differentiation to simplify the expression and find its derivative. The derivative of the given expression is:
f'(x) = (-x2ex - 2xe^x - e^x) / (x + ex)2
This result can be used to solve a variety of problems in calculus and other fields of mathematics.
References
- [1] Stewart, J. (2016). Calculus: Early Transcendentals. Cengage Learning.
- [2] Anton, H. (2017). Calculus: A New Horizon. John Wiley & Sons.
- [3] Edwards, C. H. (2018). Calculus. Pearson Education.
Glossary
- Derivative: A measure of how a function changes as its input changes.
- Quotient Rule: A rule in calculus that helps us find the derivative of a quotient of two functions.
- Chain Rule: A rule in calculus that helps us find the derivative of a composite function.
- Product Rule: A rule in calculus that helps us find the derivative of a product of two functions.
- Sum Rule: A rule in calculus that helps us find the derivative of a sum of two functions.
Derivative of a Complex Expression: Q&A =============================================
Introduction
In our previous article, we found the derivative of a complex expression with respect to a variable x. We used the quotient rule and chain rule of differentiation to simplify the expression and find its derivative. In this article, we will answer some frequently asked questions related to the derivative of a complex expression.
Q: What is the derivative of a complex expression?
A: The derivative of a complex expression is a measure of how the expression changes as its input changes. It is a fundamental concept in calculus that helps us understand how functions behave.
Q: How do I find the derivative of a complex expression?
A: To find the derivative of a complex expression, you can use the quotient rule and chain rule of differentiation. The quotient rule states that if you have a function of the form:
f(x) = g(x) / h(x)
where g(x) and h(x) are both functions of x, then the derivative of f(x) with respect to x is given by:
f'(x) = (h(x)g'(x) - g(x)h'(x)) / h(x)^2
The chain rule states that if you have a function of the form:
f(x) = g(h(x))
where g(x) and h(x) are both functions of x, then the derivative of f(x) with respect to x is given by:
f'(x) = g'(h(x)) * h'(x)
Q: What is the quotient rule?
A: The quotient rule is a rule in calculus that helps us find the derivative of a quotient of two functions. It states that if you have a function of the form:
f(x) = g(x) / h(x)
where g(x) and h(x) are both functions of x, then the derivative of f(x) with respect to x is given by:
f'(x) = (h(x)g'(x) - g(x)h'(x)) / h(x)^2
Q: What is the chain rule?
A: The chain rule is a rule in calculus that helps us find the derivative of a composite function. It states that if you have a function of the form:
f(x) = g(h(x))
where g(x) and h(x) are both functions of x, then the derivative of f(x) with respect to x is given by:
f'(x) = g'(h(x)) * h'(x)
Q: How do I apply the quotient rule and chain rule?
A: To apply the quotient rule and chain rule, you need to identify the functions g(x) and h(x) in the given expression. Then, you need to find the derivatives of g(x) and h(x) with respect to x. Finally, you can use the quotient rule and chain rule to find the derivative of the given expression.
Q: What are some common mistakes to avoid when finding the derivative of a complex expression?
A: Some common mistakes to avoid when finding the derivative of a complex expression include:
- Not identifying the functions g(x) and h(x) correctly
- Not finding the derivatives of g(x) and h(x) correctly
- Not applying the quotient rule and chain rule correctly
- Not simplifying the expression correctly
Q: How do I simplify the expression after finding the derivative?
A: To simplify the expression after finding the derivative, you can use algebraic manipulations such as combining like terms, factoring, and canceling out common factors.
Conclusion
In this article, we have answered some frequently asked questions related to the derivative of a complex expression. We have discussed the quotient rule and chain rule of differentiation, and provided examples of how to apply these rules. We have also discussed some common mistakes to avoid when finding the derivative of a complex expression. By following these guidelines, you can find the derivative of a complex expression with confidence.
References
- [1] Stewart, J. (2016). Calculus: Early Transcendentals. Cengage Learning.
- [2] Anton, H. (2017). Calculus: A New Horizon. John Wiley & Sons.
- [3] Edwards, C. H. (2018). Calculus. Pearson Education.
Glossary
- Derivative: A measure of how a function changes as its input changes.
- Quotient Rule: A rule in calculus that helps us find the derivative of a quotient of two functions.
- Chain Rule: A rule in calculus that helps us find the derivative of a composite function.
- Product Rule: A rule in calculus that helps us find the derivative of a product of two functions.
- Sum Rule: A rule in calculus that helps us find the derivative of a sum of two functions.