To find the derivative of \( \frac{1}{\log(x)} \) with respect to \( x \), we can use the chain rule. Here's the process:

### Given Function:
\[ y = \frac{1}{\log(x)} \]

### Step 1: Rewrite the function
The function can be rewritten as:
\[ y = (\log(x))^{-1} \]

### Step 2: Apply the chain rule
Using the chain rule, the derivative of \( y \) with respect to \( x \) is:
\[ \frac{dy}{dx} = -(\log(x))^{-2} \cdot \frac{d}{dx}[\log(x)] \]

### Step 3: Differentiate \( \log(x) \)
The derivative of \( \log(x) \) with respect to \( x \) is:
\[ \frac{d}{dx}[\log(x)] = \frac{1}{x} \]

### Step 4: Combine the results
Now, substitute the derivative of \( \log(x) \) back into the expression:
\[ \frac{dy}{dx} = -\frac{1}{(\log(x))^2} \cdot \frac{1}{x} \]

### Final Answer:
\[ \frac{dy}{dx} = -\frac{1}{x (\log(x))^2} \]

Would you like more details or have any questions?

Here are 8 related questions you might consider:

1. Do you want to explore more derivatives involving logarithms?
2. Would you like to see an example of applying the product or quotient rule?
3. Are you interested in integrating functions involving logarithms?
4. Do you need help understanding the chain rule in more depth?
5. Would you like to explore higher-order derivatives of this function?
6. Do you want to see how this derivative is applied in real-world problems?
7. Are there any other functions you'd like to differentiate?
8. Would you like a refresher on the basic rules of differentiation?

**Tip:** When working with logarithmic functions, always remember the domain restrictions, especially when differentiating!

Learn how to find the derivative of 1 over log(x) with a detailed step-by-step solution using differentiation and the chain rule. This problem covers differentiation of logarithmic functions and is suitable for advanced high school students and above.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation