AP Calculus AB and BC glossary

Differentiable function

A function is differentiable on an open interval when its derivative exists at every point of it, and on a closed interval when it exists at every interior point and as a one-sided derivative at each endpoint. Polynomials, sine, cosine, and e to the x qualify everywhere.

Differentiability is defined one point at a time and then extended to a whole set. A function is differentiable on an open interval when the defining limit returns a finite number at every $x$ in it, and on a closed interval $[a, b]$ when it is differentiable at every interior point and the appropriate one-sided version of that limit exists at each endpoint.

f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}

Several families are differentiable at every point of their domain: polynomials, $\sin x$ , $\cos x$ , and $e^{x}$ on all real numbers, and $\ln x$ on $x > 0$ . Differentiability on an open interval, paired with continuity on the closed one, supplies the smoothness hypotheses behind the Mean Value Theorem and Rolle's Theorem, so a question that hides a corner inside a piecewise definition is really checking whether you tested the hypothesis instead of assuming it.

The mistake

Judging an interval of differentiability from how tidy the formula looks. The function $f(x) = x^{1/3}$ is defined for every real number and is written as one clean expression, but $f'(x) = \frac{1}{3}x^{-2/3}$ is undefined at $x = 0$ , where the graph has a vertical tangent. No interval containing 0 is an interval of differentiability.

Appears in: Unit 2: Defining the Derivative