I've recently started learning calculus, and I'm trying to grasp the concept of derivatives. In my textbook, two seemingly different definitions of the derivative are presented:
- The first definition states:
$$ \frac{d\ f(x)}{dx} = \lim_{h \to 0} \frac {f(x+h) - f(x)}{h}, \text{ provided limit exists} \implies f'(c) = \lim_{h \to 0} \frac {f(c+h) - f(c)}{h}, \text{ provided limit exists} $$
- Later in the book, another expression is used:
$$ f'(c) = \lim_{x \to c} \frac {f(x) - f(c)}{x-c}, \text{ provided limit exists} $$
I'm struggling to see how these two definitions represent the same concept. Could someone help clarify this for me?
Thank you in advance, and I apologize if this question has been asked before or if there are any formatting errors.
(1.) expresses the definition of the derivative as the difference between $f(x+h)$ and $f(x)$ divided by the difference between $x$ and $x+h$ as approaches $h\to 0$, where $h$ is the difference between the points you are trying to find the slope of, which becomes $0$ for instantaneous rates of change.
(2.) approaches this in a different way. Instead of letting the difference approach $0$, $x$ approaches the point at which the derivative is calculated, which is another way of saying the difference between $x$ and the point is $0$. This is a way of saying as two different points become the same point, and you have to find the "slope" between these points. When two points near each other, the difference between them approaches $0$. This is equivalent to $h\to 0$ in (1.). By @Ted's comment, substituting $x=c+h$ changes the limit from $x\to c$ into $h\to 0$, and the rest of (2.) turns into (1.)