AP Calculus AB and BC glossary
Derivative at a point
The derivative at a point is the limit of the difference quotient as the interval shrinks to zero. It is a single number: the slope of the tangent line to the graph at that point, and the instantaneous rate of change of the function there.
Distinguish it from the derivative function. The derivative at a point is a number attached to one input; the derivative function is a rule that produces that number for every input where it exists.
The mistake
Substituting the point before differentiating. Plugging a into f and then differentiating gives zero every time, because you have differentiated a constant.
Appears in: Unit 2: Defining the Derivative