So y + Δy = f(x + Δx)
Increment on the function is given by
Δy = f(x + Δx) - f(x)
Increment of the function with respect to x is given by
Δy/Δx =[ f(x + Δx) - f(x)]/Δx
f'(x) is the limit of the above expression as Δx tends to zero.
Interpretation of f'(x) depends on the nature of the function. If function represents a curve, f'(x) is the slope at a given point.