Differentiate sin inverse x + lnroot 1-x²

Let's differentiate the function \( f(x) = \sin^{-1}(x) + \ln(\sqrt{1 - x^2}) \).

### Step-by-step differentiation:

1. **Differentiate \( \sin^{-1}(x) \):**
   \[
   \frac{d}{dx} \left( \sin^{-1}(x) \right) = \frac{1}{\sqrt{1 - x^2}}
   \]

2. **Simplify \( \ln(\sqrt{1 - x^2}) \):**

   \[
   \ln(\sqrt{1 - x^2}) = \frac{1}{2} \ln(1 - x^2)
   \]
   
   Now, differentiate \( \frac{1}{2} \ln(1 - x^2) \) using the chain rule:
   \[
   \frac{d}{dx} \left( \frac{1}{2} \ln(1 - x^2) \right) = \frac{1}{2} \cdot \frac{1}{1 - x^2} \cdot (-2x) = \frac{-x}{1 - x^2}
   \]

### Final derivative:
Combining both parts, the derivative of \( f(x) \) is:
\[
f'(x) = \frac{1}{\sqrt{1 - x^2}} - \frac{x}{1 - x^2}
\]

This is the derivative of the given function.

Do you have any questions or would you like more details?

Here are 5 questions to explore further:
1. How would you differentiate other inverse trigonometric functions?
2. What are the conditions for the derivative of \( \sin^{-1}(x) \) to exist?
3. How does the chain rule apply to logarithmic differentiation?
4. How do you differentiate products and quotients of functions?
5. What happens if you differentiate an inverse hyperbolic function?

**Tip:** When differentiating logarithmic functions, always simplify them as much as possible first for easier application of rules.

This page provides a detailed solution to differentiate sin inverse x + ln(√(1 - x²)). It covers key concepts such as inverse trigonometric functions, logarithmic differentiation, and the chain rule. Perfect for students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation