explain:im​−*hf*(*x*)−*f*(*x*+*h*)​=*h*→0lim​*hf*(*x*−*h*)−*f*(*x*)​

The expression you provided is related to the **definition of the derivative** of a function \( f(x) \) at a point \( x \), using two different forms of the difference quotient.

Here’s the breakdown:

1. **Expression 1**:  
   \[
   \lim_{h \to 0} \frac{f(x) - f(x+h)}{h}
   \]
   This is a form of the **difference quotient**. Usually, the derivative is given by:
   \[
   f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}
   \]
   But notice that in your expression, the terms are **reversed**: \( f(x) - f(x+h) \) instead of \( f(x+h) - f(x) \). This gives the **negative** of the derivative, so this is:
   \[
   \lim_{h \to 0} \frac{f(x) - f(x+h)}{h} = -f'(x)
   \]

2. **Expression 2**:  
   \[
   \lim_{h \to 0} \frac{f(x-h) - f(x)}{h}
   \]
   This is another form of the **difference quotient**, but this time it’s a **backward difference** (looking at values of \( f(x) \) to the left of \( x \), i.e., \( f(x-h) \) instead of \( f(x+h) \)). It’s also equivalent to the derivative of \( f(x) \), so:
   \[
   \lim_{h \to 0} \frac{f(x-h) - f(x)}{h} = f'(x)
   \]

### Summary:

- The first limit represents \( -f'(x) \), the **negative derivative** of \( f(x) \).
- The second limit represents \( f'(x) \), the **derivative** of \( f(x) \).

So, in essence:
\[
\lim_{h \to 0} \frac{f(x) - f(x+h)}{h} = -f'(x), \quad \lim_{h \to 0} \frac{f(x-h) - f(x)}{h} = f'(x)
\]

### Do you need further details or have any questions?

---

Here are 5 related questions to deepen your understanding:
1. What is the geometric interpretation of the derivative?
2. How does the backward difference quotient differ from the forward difference quotient?
3. Can you find the derivative of \( f(x) = x^2 \) using both difference quotients?
4. What does it mean when the derivative of a function is zero?
5. How does the derivative relate to the slope of a tangent line?

### Tip:  
Always pay attention to the direction of the difference quotient when calculating derivatives. Forward and backward difference quotients may seem similar but can yield different signs!

explain:lim_{h→0} (f(x)−f(x+h))/h − (f(x−h)−f(x))/h

Explore the difference quotient method for finding the derivative of a function. This page covers both forward and backward difference quotients, explains the mathematical formulas, and shows how they relate to the derivative. Perfect for undergraduate calculus students.

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