https://file.aiquickdraw.com/mathbot/file/1729926965845ftop8l3o.jpg

To solve this problem, let’s start by finding the derivative of the given function.

### Given Function
The function provided is:
\[
y = \ln(5 \tan^{-1} x)
\]

We are asked to find the expression for \( y' \tan^{-1} x \).

### Step 1: Differentiate \( y = \ln(5 \tan^{-1} x) \)
Using the chain rule, the derivative \( y' \) is:
\[
y' = \frac{d}{dx} \ln(5 \tan^{-1} x) = \frac{1}{5 \tan^{-1} x} \cdot \frac{d}{dx}(5 \tan^{-1} x)
\]

Now, since \( \frac{d}{dx}(\tan^{-1} x) = \frac{1}{1 + x^2} \):
\[
\frac{d}{dx}(5 \tan^{-1} x) = 5 \cdot \frac{1}{1 + x^2} = \frac{5}{1 + x^2}
\]

Thus,
\[
y' = \frac{1}{5 \tan^{-1} x} \cdot \frac{5}{1 + x^2} = \frac{1}{\tan^{-1} x} \cdot \frac{1}{1 + x^2}
\]

### Step 2: Multiply by \( \tan^{-1} x \)
Now, we are asked to find \( y' \tan^{-1} x \):
\[
y' \tan^{-1} x = \frac{1}{\tan^{-1} x} \cdot \frac{1}{1 + x^2} \cdot \tan^{-1} x = \frac{1}{1 + x^2}
\]

### Conclusion
The answer is:
\[
\frac{1}{1 + x^2}
\]

### Correct Option
The correct answer from the given choices is:
\[
\frac{1}{1 + x^2}
\]

Would you like further details or have any questions?

Here are five related questions to deepen your understanding:
1. How do you apply the chain rule in derivatives involving logarithmic functions?
2. What is the derivative of \( \tan^{-1} x \), and how is it used in this problem?
3. How do you find derivatives of compositions involving both trigonometric and logarithmic functions?
4. Why does the chain rule apply specifically when differentiating nested functions?
5. How would the solution change if the function were \( \ln(\tan^{-1} x) \) without the constant multiplier?

**Tip:** Remember that logarithmic differentiation is useful for functions involving products and powers, as it can simplify complex expressions.

If y = ln(5 tan⁻¹x), then y' * tan⁻¹x = ?

Find the derivative of y = ln(5 tan⁻¹x) and calculate y' * tan⁻¹x. This problem demonstrates the application of the chain rule and logarithmic differentiation for composite trigonometric functions. Suitable for Grade 12 or undergraduate calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation