Short on time? This page is the recap. For explanations, the graph and worked examples, use the full study guide. This topic is shared by Calculus AB and Calculus BC.
Notation: f′(x), dy/dx and df/dx all name the same derivative. dV/dt is the rate of change of V with respect to t.
Recap
- f′(a) is the instantaneous rate of change of f with respect to its input, at the input value a.
- Sign: f′(a) > 0 means f is increasing there; f′(a) < 0 means decreasing; f′(a) = 0 means momentarily not changing.
- Units: (units of f) ÷ (units of input). Second derivative: (units of f) ÷ (units of input)².
- The input is often time, but it can be anything: price, distance, altitude, quantity.
- From a table, estimate f′(a) by an average rate of change over the closest interval around a.
Key relationships
| Quantity | Meaning | Example units (W in litres, t in minutes) |
|---|---|---|
| W(a) | The amount at input a | litres |
| (W(b) − W(a))/(b − a) | Average rate over [a, b] | litres per minute |
| W′(a) | Rate at the single input a | litres per minute |
| W″(a) | How fast the rate W′ is changing at a | litres per minute per minute |
| W′(a) × Δt | Approximate change in W over a short step Δt | litres |
Four-part interpretation sentence: quantity + input value (with units) + increasing/decreasing + rate with units. “At t = 10 minutes, the amount of water in the tank is increasing at a rate of 3 litres per minute.”
Assumptions behind the method
- The function is differentiable at the input value you are interpreting.
- A table only gives an estimate of the derivative. Use the closest data points on either side.
- “Change ≈ rate × step” works only for small steps, because the rate itself may change.
Mistakes to avoid
- Writing f(a) when you mean f′(a) (an amount instead of a rate).
- Units that are a product, or no units at all.
- “Decreasing at a rate of −5”: a double negative. Drop the minus sign after “decreasing”.
- Claiming f′(a) is the exact change over the next unit of input.
- Leaving out the input value (“at t = 10 minutes”).
- Mentioning time when the input is something else, such as price.
Quick self-check
- h(t) is the height of a plant in centimetres, t in days, and h′(10) = 0.8. Interpret, then give the rate in millimetres per day. (On day 10 the plant is growing at 0.8 cm per day; that is 8 mm per day.)
- P(x) is profit in dollars from selling x items. What are the units of P′(x)? Of P″(x)? (Dollars per item; dollars per item per item.)
- W(t) is the water in a tank in litres, t in hours. The tank holds 500 litres at t = 4 and W′(4) = −20. Is the tank empty soon after t = 4? (No. It is losing water at 20 litres per hour at that instant, but it still holds about 500 litres.)
Next: practice questions.