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 dy dx = lim h0 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=Δx

and

f(x+h)f(x)= y 1 y=Δy

where Δx  represents the change in x  and Δy  represents the change in y .

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

Thus we have

dy dx = lim h0 f(x+h)f(x) h = lim Δx0 Δy Δx