### Revisiting definition of derivative

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

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

If we write ${x}_{1}=x+h$  and ${y}_{1}=f\left(x+h\right)$  , then

$h=\left(x+h\right)-\left(x\right)={x}_{1}-x=\Delta x$

and

$f\left(x+h\right)-f\left(x\right)={y}_{1}-y=\Delta y$

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

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