Defining the Derivative of a Function and Using Derivative Notation AB & BC
▶︎ Watch it animatedinteractive step-through · ~3 min · optionalLeaving the point free turns the derivative at a point into a derivative function, $f'(x) = \lim_{h \to 0}\frac{f(x+h) - f(x)}{h}$, whose output at each input is the slope of the tangent there. Computing it takes the same four steps every time: expand, subtract, cancel the $h$, take the limit. Once $f'$ is known, any $f'(a)$ follows by substitution, and the domain of $f'$ can be smaller than the domain of $f$.
Two things go wrong. The quotient itself gets malformed, with a denominator that is an input rather than a change, a limit variable that does not match, or the $h$ left uncancelled so the limit is taken too early. And the notation gets confused: answering with $f'(x)$ when a number $f'(a)$ was asked for, treating $\frac{dy}{dx}$ as a fraction to split, or reading $\frac{d}{dx}[f(x)]$ at a point as a value of $f$ rather than a slope.
The work
3 ways in · any order
Lesson
Defining the Derivative of a Function and Using Derivative Notation
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Turns the rate at a single point into a derivative function, runs the four-step computation from the definition on a polynomial and a reciprocal, and sorts out which notations name a function and which name a number at a point.
Diagnostic
10-item topic check
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Ten items spanning the two failure modes of this topic: a difference quotient written with the wrong increment, denominator or limit, and notation confusion between the derivative function and its value at a point.
Targeted Practice
Drill a single misconception
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Pick one of the failure modes you missed and drill it on its own. The round is adaptive: two correct in a row clears it for now and moves you to the next. Two in a row is a checkpoint, not proof: if the error resurfaces later, the misconception comes back.