AP Calculus AB and BC glossary

Derivative of a constant

Also called: Constant rule

The derivative of any constant is 0, because the graph of a constant function is a horizontal line and a horizontal line has slope 0. Differentiation erases the constant term, which is why every antiderivative carries a constant of integration to record what was lost.

The difference quotient settles it before any limit work happens. For $f(x) = c$ , every numerator $f(x+h) - f(x)$ equals $c - c = 0$ , so the quotient is already 0 for each $h \ne 0$ and the limit has nothing left to do.

f(x) = c \quad \Longrightarrow \quad f'(x) = \lim_{h \to 0} \frac{c - c}{h} = 0

The consequence runs both ways. Two functions differing by a constant have identical derivatives, so a derivative cannot tell $x^{2}$ from $x^{2} + 7$ , and on an interval the Mean Value Theorem gives the converse. Reversing the process therefore returns a whole family $F(x) + C$ rather than a single function, and an initial condition is what selects the one member the problem wants.

The mistake

Deciding what counts as a constant by hunting for digits. Symbols such as $\pi$ , $e^{2}$ , and $\ln 5$ are just numbers, so each has derivative 0, yet $e^{2}$ gets differentiated as $2e$ every year. The opposite slip is as common: the derivative of $kx$ with $k$ constant is $k$ , not 0, because there the constant multiplies a variable instead of standing alone.

Appears in: Unit 2: Defining the Derivative