The Definition of Derivative

Discard the Instantaneous Changing Rate

It is common that the derivative is the instantaneous changing rate. However, it is actually wrong since the rate of change needs a period of time to be meaningful.

When we talk about the derivative, we first imply the average changing rate, which describes how ${\displaystyle f(x)}$ changes when ${\displaystyle x}$ differs, that is:

${\displaystyle \frac{\mathrm{\Delta }f(x)}{\mathrm{\Delta }x}}$

Suppose when ${\displaystyle \mathrm{\Delta }x=0}$, the ratio becomes the form ${\displaystyle \frac{0}{0}}$, which have no meaning. So we can not just simply consider derivative as the instantaneous rate of changing.

Proof By Limits

We now know that the derivative is not the instantaneous changing rate, but what it is?

First we transform ${\displaystyle \frac{\mathrm{\Delta }f(x)}{\mathrm{\Delta }x}}$ into a more common form:

${\displaystyle \frac{f(x+h)-f(x)}{h}}$

where ${\displaystyle h}$ is actually ${\displaystyle \mathrm{\Delta }x}$.

When we have ${\displaystyle h\to 0}$, it is easy to be confused that ${\displaystyle \frac{f(x+h)-f(x)}{h}}$ is infinitely small. But, actually, it is not. In the opposite way, when the ratio is a infinitesimal, the derivative should not exists.

In the case of limit, we have:

${\displaystyle f^{\prime }(x)=\underset{h\to o}{lim}\frac{f(x+h)-f(x)}{h}}$

Take ${\displaystyle f(x)=x^2}$ as an example, when we expand and simplify the expression:

${\displaystyle \frac{\mathrm{\Delta }f}{h}=\frac{(x+h{)}^2-x^2}{h}=2x+h}$

When ${\displaystyle h\to 0}$, take the equation as the form of limit, the ratio is actually ${\displaystyle 2x}$ which is not ${\displaystyle 0}$.

This case exist if and only if the limit exists, and the ratio of two infinitesimals is what we say derivative.

So, we have the definition of derivative expressed by limit:

${\displaystyle f^{\prime }(x)=\underset{h\to o}{lim}\frac{f(x+h)-f(x)}{h}}$

It is not the instantaneous rate of changing, but the ratio of two infinitesimals(Not so exactly, actually the coefficient of two differentials ${\displaystyle dy}$ and ${\displaystyle dx}$). It is the best approximation of the rate of change.