https://file.aiquickdraw.com/mathbot/file/1727857456347zk9w5zs7.jpg

The image shows the differentiation of the function \( \frac{7}{x} \). Here's the explanation step-by-step:

1. **Starting Function**:  
   The function is \( \frac{7}{x} \), which can be written as \( 7x^{-1} \) using the properties of exponents.

2. **Differentiate Using Power Rule**:  
   The power rule states that if \( f(x) = ax^n \), then \( f'(x) = a \cdot n \cdot x^{n-1} \).

   - Here, \( a = 7 \), \( n = -1 \).
   - The derivative becomes:
   \[
   \frac{d}{dx}(7x^{-1}) = 7 \cdot (-1) \cdot x^{-2} = -7x^{-2}.
   \]

3. **Final Answer**:  
   The result of the derivative is:
   \[
   \frac{d}{dx} \left( \frac{7}{x} \right) = -7x^{-2}.
   \]

This matches the equation shown in the image:
\[
\left( \frac{7}{x} \right)' = 7 \cdot (-1) \cdot x^{-2} = -7x^{-2}.
\]

Would you like more details or have any questions?

Here are some related questions to deepen your understanding:

1. How would you differentiate \( \frac{a}{x} \) where \( a \) is any constant?
2. How does the power rule apply to positive powers of \( x \)?
3. Can you explain why the exponent decreases by 1 in the power rule?
4. How would you apply the product rule to a more complex function?
5. What is the derivative of \( \frac{7}{x^2} \)?

**Tip**: When differentiating fractions, it often helps to rewrite them using negative exponents.

Learn how to differentiate the function 7/x step-by-step using the power rule. This problem demonstrates the application of the rule for negative exponents, suitable for students studying calculus or advanced algebra.

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