find the first and second derivative of ln(4-x^2)

To find the first and second derivatives of \( f(x) = \ln(4 - x^2) \), we'll start with the first derivative.

### First Derivative

Using the chain rule:

\[
f'(x) = \frac{d}{dx} \ln(4 - x^2) = \frac{1}{4 - x^2} \cdot \frac{d}{dx}(4 - x^2)
\]

Calculating the derivative of \( 4 - x^2 \):

\[
\frac{d}{dx}(4 - x^2) = -2x
\]

Now substituting back:

\[
f'(x) = \frac{-2x}{4 - x^2}
\]

### Second Derivative

To find the second derivative \( f''(x) \), we differentiate \( f'(x) \):

\[
f''(x) = \frac{d}{dx}\left(\frac{-2x}{4 - x^2}\right)
\]

Using the quotient rule, where \( u = -2x \) and \( v = 4 - x^2 \):

\[
f''(x) = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2}
\]

Calculating \( \frac{du}{dx} = -2 \) and \( \frac{dv}{dx} = -2x \):

Substituting into the quotient rule:

\[
f''(x) = \frac{(4 - x^2)(-2) - (-2x)(-2x)}{(4 - x^2)^2}
\]

Simplifying the numerator:

\[
f''(x) = \frac{-2(4 - x^2) - 4x^2}{(4 - x^2)^2} = \frac{-8 + 2x^2 - 4x^2}{(4 - x^2)^2} = \frac{-8 - 2x^2}{(4 - x^2)^2}
\]

### Summary of Derivatives

1. **First Derivative**: 
   \[
   f'(x) = \frac{-2x}{4 - x^2}
   \]

2. **Second Derivative**: 
   \[
   f''(x) = \frac{-8 - 2x^2}{(4 - x^2)^2}
   \]

If you have any questions or need further details, let me know! Here are some related questions you might consider:

1. What are the critical points of \( f(x) \)?
2. How do the first and second derivatives help in determining concavity?
3. What is the behavior of \( f(x) \) as \( x \) approaches the boundaries of its domain?
4. Can you graph \( f(x) \) and its derivatives?
5. How do you apply the second derivative test for extrema? 

**Tip**: Always check the domain of the function before differentiating to ensure you're working within valid intervals.

Explore the first and second derivatives of the function ln(4 - x^2) with detailed step-by-step solutions. This guide covers important concepts in calculus, including the application of the chain rule and quotient rule. Suitable for advanced high school 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