Remark  Functions with same derivatives differ by a constant To show this, let g and h
be two functions having the same derivatives on an interval I Consider the function f = g – h defined by f (x) = g(x) – h(x), ∀ x ∈ I
Then
df
dx = f′ = g′  – h′ giving  f′ (x) = g′ (x) – h′ (x) ∀ x ∈ I
or
f′ (x) = 0, ∀x ∈ I by hypothesis,
i e