# Real Analysis/Exercises on Derivatives

## On Differentiation

The Calculus wikibook contains many questions that simply practice the differentiation rules. It may cover differentiation rules that we have not covered, but that is because we have yet to define them right now.

## On Derivatives

1. Prove that for any even function ${\displaystyle f}$  provided that ${\displaystyle f'(0)}$  is valid, ${\displaystyle f'(0)}$  must equal 0.

Question Hint

Given a function ƒ such that ƒ(x) = ƒ(-x) ∀x, prove that if ƒ'(0) is valid, ƒ'(0) = 0.

Methodology Hint 1a

It is easy to prove this using contradiction (${\displaystyle P\land \neg Q{\text{ but }}P\rightarrow Q}$ ).

Methodology Hint 1b

P in this case is "the function is even". Q in this case is "ƒ'(0) = 0". From there, prove that -Q implies -P, to highlight the contradiction.