Revisiting definition of derivative

Let us revisit the definition of the derivative and try to see it in a different way.

Let $y=f(x)$  , then $\frac{dy}{dx}=\underset{h\to 0}{\lim }\frac{f(x+h)-f(x)}{h}$

If we write $x_1=x+h$  and $y_1=f(x+h)$  , then

$h=(x+h)-(x)=x_1-x=\Delta x$

and

$f(x+h)-f(x)=y_1-y=\Delta y$

where $\Delta x$  represents the change in $x$  and $\Delta y$  represents the change in $y$.

The notation $\Delta x$  or $\Delta y$  should not be confused with a product. The Greek capital letter $\Delta$  is read as 'delta' and is used in place of the letter 'd'. The choice of the letter is convenient as $\Delta x$  represents the change or difference in $x$  values and so is $\Delta y$ the change or difference in $y$.

Thus we have

$\frac{dy}{dx}=\underset{h\to 0}{\lim }\frac{f(x+h)-f(x)}{h}=\underset{\Delta x\to 0}{\lim }\frac{\Delta y}{\Delta x}$