Problem 33
Question
In Exercises \(21-42,\) find the derivative of \(y\) with respect to the appropriate variable. $$ y=\ln \left(\tan ^{-1} x\right) $$
Step-by-Step Solution
Verified Answer
The derivative of \( y = \ln(\tan^{-1}x) \) is \( \frac{1}{\tan^{-1}x \cdot (1+x^2)} \).
1Step 1: Identify the Function's Components
The function given is \( y = \ln(\tan^{-1}x) \). This means we have a composite function where the outer function is a natural logarithm \( \ln \) and the inner function is the arctangent \( \tan^{-1}x \). We will apply the chain rule to find its derivative.
2Step 2: Differentiate the Outer Function
By the chain rule, the outer function \( \ln(u) \), where \( u = \tan^{-1}x \), has a derivative of \( \frac{1}{u} \cdot \frac{du}{dx} \). Write it as \( \frac{1}{\tan^{-1}x} \cdot \frac{d}{dx}(\tan^{-1}x) \).
3Step 3: Differentiate the Inner Function
The derivative of \( \tan^{-1}x \) with respect to \( x \) is \( \frac{1}{1+x^2} \). Substitute this result back into our expression from Step 2.
4Step 4: Combine the Results
Substitute the derivative of the inner function into the expression from Step 2 to get \( \frac{1}{\tan^{-1}x} \cdot \frac{1}{1+x^2} \). This gives us the final derivative \( \frac{1}{\tan^{-1}x \cdot (1+x^2)} \).
Key Concepts
Chain RuleNatural LogarithmArctangent Function
Chain Rule
The chain rule is a crucial concept in calculus used to differentiate composite functions. A composite function, in simple terms, is a function within another function.
For example, in the function \( y = \ln(\tan^{-1}x) \), the arctangent function \( \tan^{-1}x \) is nestled within the natural logarithm \( \ln \).
To differentiate composite functions, the chain rule tells us to differentiate the outer function and then multiply by the derivative of the inner function. This process allows us to break down complex functions into more manageable parts. The general form of the chain rule is:
\[\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}\]
Here, \( \frac{dy}{du} \) represents the derivative of the outer function, and \( \frac{du}{dx} \) represents the derivative of the inner function.
When applying this rule to our example:
For example, in the function \( y = \ln(\tan^{-1}x) \), the arctangent function \( \tan^{-1}x \) is nestled within the natural logarithm \( \ln \).
To differentiate composite functions, the chain rule tells us to differentiate the outer function and then multiply by the derivative of the inner function. This process allows us to break down complex functions into more manageable parts. The general form of the chain rule is:
\[\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}\]
Here, \( \frac{dy}{du} \) represents the derivative of the outer function, and \( \frac{du}{dx} \) represents the derivative of the inner function.
When applying this rule to our example:
- Start by identifying \( u = \tan^{-1}x \) as the inner function.
- The outer function is \( \ln(u) \).
- Differentiating \( \ln(u) \) gives \( \frac{1}{u} \cdot \frac{du}{dx} \).
Natural Logarithm
The natural logarithm, denoted as \( \ln(x) \), is a fundamental logarithmic function in calculus and can be defined for any positive number \( x \).
It is the inverse of the exponential function \( e^x \). The base \( e \) (approximately 2.718) is a unique number known as Euler's number.
The importance of the natural logarithm arises from its properties and derivatives. Specifically, the derivative of \( \ln(x) \) is straightforward:
\[\frac{d}{dx}[\ln(x)] = \frac{1}{x}\]
This property helps simplify various calculations in calculus and physics. In our context, when the natural logarithm operates on \( \tan^{-1}x \), we treat \( \tan^{-1}x \) as "\( x \)", leading to the differentiation process we saw.
It is the inverse of the exponential function \( e^x \). The base \( e \) (approximately 2.718) is a unique number known as Euler's number.
The importance of the natural logarithm arises from its properties and derivatives. Specifically, the derivative of \( \ln(x) \) is straightforward:
\[\frac{d}{dx}[\ln(x)] = \frac{1}{x}\]
This property helps simplify various calculations in calculus and physics. In our context, when the natural logarithm operates on \( \tan^{-1}x \), we treat \( \tan^{-1}x \) as "\( x \)", leading to the differentiation process we saw.
- This means, when differentiating \( \ln(\tan^{-1}x) \), the derivative of \( \ln \) contributes \( \frac{1}{\tan^{-1}x} \).
- The subsequent step involves leveraging the chain rule to differentiate \( \tan^{-1}x \).
Arctangent Function
The arctangent function, represented as \( \tan^{-1}x \), is the inverse function of the tangent function \( \tan(x) \).
This means that it works in the opposite way, converting a given value back into an angle.
Its derivative is a key component in calculus, primarily because it shows up often due to its relation with trigonometric identities.
For the function \( \tan^{-1}x \), the derivative with respect to \( x \) is:
\[\frac{d}{dx}[\tan^{-1}x] = \frac{1}{1+x^2}\]
This result comes from understanding how changes in \( x \) affect the output of the arctangent. It is particularly useful when differentiating functions involving inverse trigonometric identities.
This means that it works in the opposite way, converting a given value back into an angle.
Its derivative is a key component in calculus, primarily because it shows up often due to its relation with trigonometric identities.
For the function \( \tan^{-1}x \), the derivative with respect to \( x \) is:
\[\frac{d}{dx}[\tan^{-1}x] = \frac{1}{1+x^2}\]
This result comes from understanding how changes in \( x \) affect the output of the arctangent. It is particularly useful when differentiating functions involving inverse trigonometric identities.
- This derivative is key in our initial problem because we need it when applying the chain rule.
- By using this derivative, the chain rule helps us conclude how the combination of \( \ln \) and \( \tan^{-1}x \) behaves.
Other exercises in this chapter
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