关于导数两种定义等价性的疑问

Hey there! No need to apologize at all—this is a super common (and totally valid) question when you're starting out with derivatives. Let's break this down step by step so it clicks.

First, let's restate the two definitions clearly to make sure we're on the same page:

1. The "h→0" definition:
   $$
   f'(c) = \lim_{h \to 0} \frac {f(c+h) - f(c)}{h}, \text{ provided limit exists}
   $$
2. The "x→c" definition:
   $$
   f'(c) = \lim_{x \to c} \frac {f(x) - f(c)}{x-c}, \text{ provided limit exists}
   $$

The key to seeing they're the same is a simple variable substitution. Let's take the second definition first: let's set $h = x - c$. That means $x = c + h$. Now, think about the limit condition: when $x$ approaches $c$, $x - c$ (which is $h$) approaches 0.

If we substitute $x = c + h$ into the second definition, we get:
$$
f'(c) = \lim_{h \to 0} \frac {f(c+h) - f(c)}{h}
$$
Which is exactly the first definition!

We can reverse this too: start with the first definition, set $x = c + h$ (so $h = x - c$). When $h$ approaches 0, $x = c + h$ approaches $c$. Substituting back gives us the second definition.

To put it in more intuitive terms: both definitions are calculating the slope of the tangent line to $y=f(x)$ at the point $(c, f(c))$.

• The first one uses a small "step" $h$ away from $c$, looking at the slope between $(c, f(c))$ and $(c+h, f(c+h))$ as that step shrinks to nothing.
• The second one looks at the slope between $(c, f(c))$ and any nearby point $(x, f(x))$, as that nearby point gets closer and closer to $c$.

They're just two different ways to describe the same idea of "zooming in" infinitely close to the point $c$ to find the tangent slope.

备注：内容来源于stack exchange，提问作者user1218456