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?

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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!