Differentiate The Following With Respect To { X $}$:21) { Y = \tan {-1}\left(\frac{x 2 + 1}{2x + 3}\right) $}$22) { Y = \tan {-1}(3x 2) $}$23) { Y = 2 \ln X $}$24) { Y = E^{\ln X} $} 25 ) \[ 25) \[ 25 ) \[ Y
Introduction
In calculus, we often encounter functions that involve inverse trigonometric and logarithmic functions. These functions are essential in various mathematical and real-world applications, such as physics, engineering, and economics. In this article, we will focus on differentiating inverse trigonometric and logarithmic functions with respect to x. We will explore five different functions and provide step-by-step solutions to each.
Function 1: y = tan(-1)((x2 + 1)/(2x + 3))
The first function we will differentiate is y = tan(-1)((x2 + 1)/(2x + 3)). To differentiate this function, we will use the chain rule and the fact that the derivative of tan^(-1)(u) is 1/(1 + u^2).
Step 1: Identify the inner function
The inner function is (x^2 + 1)/(2x + 3).
Step 2: Differentiate the inner function
Using the quotient rule, we get:
d/dx ((x^2 + 1)/(2x + 3)) = ((2x + 3)(2x) - (x^2 + 1)(2)) / (2x + 3)^2
Step 3: Simplify the derivative
Simplifying the derivative, we get:
d/dx ((x^2 + 1)/(2x + 3)) = (4x^2 + 6x - 2x^2 - 2) / (2x + 3)^2
Step 4: Differentiate the outer function
The outer function is tan^(-1)(u), where u = (x^2 + 1)/(2x + 3). Using the chain rule, we get:
d/dx tan(-1)((x2 + 1)/(2x + 3)) = 1 / (1 + ((x^2 + 1)/(2x + 3))^2) * d/dx ((x^2 + 1)/(2x + 3))
Step 5: Simplify the derivative
Simplifying the derivative, we get:
d/dx tan(-1)((x2 + 1)/(2x + 3)) = (2x + 3)^2 / (4x^2 + 6x - 2x^2 - 2)^2
Function 2: y = tan(-1)(3x2)
The second function we will differentiate is y = tan(-1)(3x2). To differentiate this function, we will use the chain rule and the fact that the derivative of tan^(-1)(u) is 1/(1 + u^2).
Step 1: Identify the inner function
The inner function is 3x^2.
Step 2: Differentiate the inner function
Using the power rule, we get:
d/dx (3x^2) = 6x
Step 3: Differentiate the outer function
The outer function is tan^(-1)(u), where u = 3x^2. Using the chain rule, we get:
d/dx tan(-1)(3x2) = 1 / (1 + (3x2)2) * d/dx (3x^2)
Step 4: Simplify the derivative
Simplifying the derivative, we get:
d/dx tan(-1)(3x2) = 6x / (1 + 9x^4)
Function 3: y = 2 ln x
The third function we will differentiate is y = 2 ln x. To differentiate this function, we will use the chain rule and the fact that the derivative of ln(x) is 1/x.
Step 1: Identify the inner function
The inner function is ln x.
Step 2: Differentiate the inner function
Using the fact that the derivative of ln(x) is 1/x, we get:
d/dx ln x = 1/x
Step 3: Differentiate the outer function
The outer function is 2u, where u = ln x. Using the chain rule, we get:
d/dx 2 ln x = 2 * d/dx ln x
Step 4: Simplify the derivative
Simplifying the derivative, we get:
d/dx 2 ln x = 2/x
Function 4: y = e^(ln x)
The fourth function we will differentiate is y = e^(ln x). To differentiate this function, we will use the chain rule and the fact that the derivative of e^u is e^u.
Step 1: Identify the inner function
The inner function is ln x.
Step 2: Differentiate the inner function
Using the fact that the derivative of ln(x) is 1/x, we get:
d/dx ln x = 1/x
Step 3: Differentiate the outer function
The outer function is e^u, where u = ln x. Using the chain rule, we get:
d/dx e^(ln x) = e^(ln x) * d/dx ln x
Step 4: Simplify the derivative
Simplifying the derivative, we get:
d/dx e^(ln x) = e^(ln x) * 1/x
Step 5: Simplify further
Simplifying further, we get:
d/dx e^(ln x) = x
Conclusion
In this article, we have differentiated four different functions involving inverse trigonometric and logarithmic functions. We have used the chain rule and the fact that the derivative of tan^(-1)(u) is 1/(1 + u^2) and the derivative of ln(x) is 1/x. We have also used the power rule and the fact that the derivative of e^u is e^u. The derivatives of the four functions are:
- d/dx tan(-1)((x2 + 1)/(2x + 3)) = (2x + 3)^2 / (4x^2 + 6x - 2x^2 - 2)^2
- d/dx tan(-1)(3x2) = 6x / (1 + 9x^4)
- d/dx 2 ln x = 2/x
- d/dx e^(ln x) = x
Introduction
In our previous article, we discussed how to differentiate inverse trigonometric and logarithmic functions. In this article, we will provide a Q&A section to help clarify any doubts and provide additional examples.
Q1: What is the derivative of tan(-1)(x2 + 1)?
A1: To find the derivative of tan(-1)(x2 + 1), we will use the chain rule and the fact that the derivative of tan^(-1)(u) is 1/(1 + u^2). The derivative is:
d/dx tan(-1)(x2 + 1) = 1 / (1 + (x^2 + 1)^2) * d/dx (x^2 + 1)
Simplifying further, we get:
d/dx tan(-1)(x2 + 1) = 2x / (1 + (x^2 + 1)^2)
Q2: How do I differentiate a function that involves both inverse trigonometric and logarithmic functions?
A2: To differentiate a function that involves both inverse trigonometric and logarithmic functions, we will use the chain rule and the fact that the derivative of tan^(-1)(u) is 1/(1 + u^2) and the derivative of ln(x) is 1/x. For example, to differentiate the function y = tan^(-1)(ln(x)), we will use the chain rule:
d/dx tan^(-1)(ln(x)) = 1 / (1 + (ln(x))^2) * d/dx ln(x)
Simplifying further, we get:
d/dx tan^(-1)(ln(x)) = 1 / (1 + (ln(x))^2) * 1/x
Q3: What is the derivative of e(ln(x2))?
A3: To find the derivative of e(ln(x2)), we will use the chain rule and the fact that the derivative of e^u is e^u. The derivative is:
d/dx e(ln(x2)) = e(ln(x2)) * d/dx ln(x^2)
Simplifying further, we get:
d/dx e(ln(x2)) = x^2 * 2/x
Q4: How do I differentiate a function that involves a composite function?
A4: To differentiate a function that involves a composite function, we will use the chain rule. For example, to differentiate the function y = sin^(-1)(cos(x)), we will use the chain rule:
d/dx sin^(-1)(cos(x)) = 1 / (1 + (cos(x))^2) * d/dx cos(x)
Simplifying further, we get:
d/dx sin^(-1)(cos(x)) = -sin(x) / (1 + (cos(x))^2)
Q5: What is the derivative of tan(-1)(ex)?
A5: To find the derivative of tan(-1)(ex), we will use the chain rule and the fact that the derivative of tan^(-1)(u) is 1/(1 + u^2). The derivative is:
d/dx tan(-1)(ex) = 1 / (1 + (ex)2) * d/dx e^x
Simplifying further, we get:
d/dx tan(-1)(ex) = e^x / (1 + (ex)2)
Conclusion
In this article, we have provided a Q&A section to help clarify any doubts and provide additional examples on differentiating inverse trigonometric and logarithmic functions. We hope that this article has been helpful in providing a clear and concise explanation of how to differentiate these types of functions.
Common Mistakes to Avoid
When differentiating inverse trigonometric and logarithmic functions, there are several common mistakes to avoid:
- Not using the chain rule when differentiating composite functions
- Not simplifying the derivative after using the chain rule
- Not using the correct derivative of the inverse trigonometric or logarithmic function
- Not checking the domain of the function before differentiating
By avoiding these common mistakes, you can ensure that you are differentiating inverse trigonometric and logarithmic functions correctly.
Practice Problems
To practice differentiating inverse trigonometric and logarithmic functions, try the following problems:
- Differentiate the function y = tan(-1)(x3 + 1)
- Differentiate the function y = ln(x^2 + 1)
- Differentiate the function y = e(ln(x2))
- Differentiate the function y = sin^(-1)(cos(x))
We hope that this article has been helpful in providing a clear and concise explanation of how to differentiate inverse trigonometric and logarithmic functions.