The first derivative, f'(x), measures the instantaneous rate of change of a function at a point, which equals the slope of the tangent line there. On the AP exam, its sign tells you where f is increasing (f' > 0) or decreasing (f' < 0), and where f' = 0 or doesn't exist marks a critical point.
The first derivative of a function f, written f'(x) or dy/dx, tells you how fast f is changing at any single point. Graphically, it's the slope of the tangent line. Verbally, it's an instantaneous rate of change, like a car's speedometer reading at one exact moment instead of the average speed over a whole trip.
The real power on the AP exam comes from reading f' as information about f. Where f'(x) > 0, the function is increasing. Where f'(x) < 0, it's decreasing. Where f'(x) = 0 or f' fails to exist, you have a critical point, which is the only place a local max or min can happen. The units of f' are always output units per input unit, so if P(t) is pandas and t is years, P'(t) is pandas per year. That units habit alone earns points on interpretation FRQs.
The first derivative is the spine of Units 4 and 5. Learning objective 5.3.A asks you to justify conclusions about a function's behavior based on its derivatives, and the standard justification is a sign analysis of f'. LO 5.2.A and FUN-1.C.2 define critical points as exactly the places where the first derivative is zero or undefined, which sets up the First and Second Derivative Tests for extrema. LO 4.3.A has you interpret f' as a rate of change in applied contexts (filling tanks, growing populations, people leaving a livestream), where you write a sentence with correct units. The concept then keeps showing up. Unit 7 builds differential equations out of derivative expressions, and on the BC side, Topics 9.1 and 9.2 extend the first derivative to parametric curves, where dy/dx = (dy/dt)/(dx/dt). If you understand the first derivative deeply, roughly a third of the course unlocks.
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Second Derivative (Unit 5)
The second derivative is just the first derivative of the first derivative. Everything f' tells you about f, f'' tells you about f'. So f'' > 0 means f' is increasing, which is what concave up actually means.
Critical Points (Unit 5)
A critical point is defined by the first derivative. It's any point where f'(x) = 0 or f' fails to exist. Per FUN-1.C.3, every local extremum lives at a critical point, but not every critical point is an extremum, so you still have to check the sign change of f'.
Accumulation Functions (Unit 6)
For g(x) = ∫ from a to x of f(t) dt, the Fundamental Theorem says g'(x) = f(x). That means the graph of f IS the first derivative of g, so all your Unit 5 skills (sign of f gives increasing/decreasing of g, zeros of f give critical points of g) transfer directly. This is the engine behind classic graph-of-f FRQs.
Derivatives of Parametric Equations (Unit 9, BC only)
When a curve is defined by x(t) and y(t), the first derivative dy/dx is still the tangent slope, but you compute it as dy/dt divided by dx/dt (as long as dx/dt ≠ 0). Same meaning, new formula.
Multiple choice loves quick interpretation. You'll get a context like a tank filling with gas or a panda population and be asked for the sign and units of the first derivative (filling tank means positive, liters per unit time; growing population means pandas per year). FRQs use the first derivative two main ways. First, interpretation in context, like the 2025 FRQ where C(t) models acres affected by an invasive species and you work with C'(t) as a rate in acres per unit time. Second, tables of function and first derivative values, like 2023 FRQ Q5, where you apply chain rule, product rule, or sign reasoning using f' values from a table. For full justification credit, you must cite the derivative explicitly. Write "f is increasing because f'(x) > 0 on the interval," not just "the graph goes up."
The first derivative describes f directly (increasing, decreasing, critical points). The second derivative describes f' (so it gives concavity and inflection points of f). The classic trap is mixing layers. f'(x) = 0 does not mean an inflection point, and f''(x) = 0 does not mean a max or min. Match each derivative to its own job, and remember the Second Derivative Test uses f'' only to classify a critical point you already found with f'.
The first derivative f'(x) gives the instantaneous rate of change of f at a point, which equals the slope of the tangent line there.
If f'(x) > 0 on an interval, f is increasing there; if f'(x) < 0, f is decreasing. This sign analysis is the standard AP justification.
A critical point occurs where the first derivative equals zero or fails to exist, and all local extrema occur at critical points (but not every critical point is an extremum).
The units of the first derivative are always output units per input unit, like liters per minute or pandas per year, and interpretation FRQs expect you to state them.
For parametric curves on the BC exam, the first derivative dy/dx equals (dy/dt)/(dx/dt) when dx/dt is not zero, and it still means tangent slope.
When g(x) is defined as an integral of f, the graph of f acts as the first derivative of g, so you analyze g's behavior using the sign of f.
It's the function f'(x) that gives the instantaneous rate of change of f at each point, equal to the slope of the tangent line. Its sign tells you where f is increasing (f' > 0) or decreasing (f' < 0).
No. f'(x) = 0 only means you have a critical point, which is a candidate for an extremum. You still need f' to change sign there (or a second derivative check) to confirm a max or min. f(x) = x³ at x = 0 is the classic counterexample.
The first derivative describes how f changes (increasing/decreasing, rate of change). The second derivative describes how f' changes, which gives you concavity and points of inflection. They answer different questions about the same graph.
Always output units divided by input units. If a tank's volume is in liters and time is in minutes, the first derivative is in liters per minute. AP free-response questions regularly award a point for stating units correctly.
Divide dy/dt by dx/dt, provided dx/dt ≠ 0. The result, dy/dx, is still the slope of the tangent line to the curve. This is a BC-only skill from Topic 9.1.