If y = f(x), y will change to y + Δy, when x changes to x + Δx

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.