Axiomatic Definition
Next we introduce a new property of real functions, differentiability.
If f is differentiable at every point in a subset S \subseteq I, we say that f is differentiable on S. If f has a derivative at c, we notate f^\prime(c).
Next we introduce a new property of real functions, differentiability.
Let I be any interval on \mathbb{R}, and let c \in I. We say that f \colon I \to \mathbb{R} is differentiable at c if
\lim_{x \to c} \frac{f(x)-f(c)}{x-c}
exists and is finite.
If f is differentiable at every point in a subset S \subseteq I, we say that f is differentiable on S. If f has a derivative at c, we notate f^\prime(c).
Let f \colon \mathbb{R} \to \mathbb{R} be a constant function. That is f(x) = d for some d \in \mathbb{R}.
Claim: The derivative of f is zero at every point c \in \mathbb{R}.
We use out formula for differentiability
f^\prime(c) = \lim_{x \to c} \frac{f(x) - f(c)}{x - c} = \lim_{x \to c} \frac{d - d}{x - c} = 0