Key takeaway: A derivative is the instantaneous rate of change of a function at a point, equal to the slope of the line tangent to its graph at that input.
Grade 12 – AP Calculus ABAP Calculus AB Unit 2Calculus
Definition of Derivative
A derivative tells how fast an output is changing at one exact input. An average rate of change uses two points and a secant line. Squeezing those two points together until only one remains gives a rate for a single instant.
Geometrically it is the slope of the tangent line. In a real setting it carries units: if depth is measured in inches and time in hours, its rate is measured in inches per hour.
Formula and notation
The derivative of \(y=f(x)\) can be written in several ways:
\[ f^{\prime}(x)=\frac{dy}{dx}=\frac{d}{dx}\,f(x). \]
The symbol \(f^{\prime}(a)\) means the derivative evaluated at \(x=a\). Read \(\frac{dy}{dx}\) as the change in \(y\) per tiny change in \(x\); it is one symbol, not a fraction to cancel.
Key facts
- A positive value means the function is increasing at that input; a negative one means decreasing.
- A value of zero marks a horizontal tangent, often at a peak or a valley.
- A function must be continuous at a point to be differentiable there, but continuity alone is not enough.
- The units are the output units divided by the input units.
Derivative example
At a Denver weather station, snow depth is modeled by \(S(t)=0.5t^2+3t\) inches, where \(t\) is the number of hours after midnight. Find \(S^{\prime}(4)\) and say what it means.
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Differentiate term by term: \(S^{\prime}(t)=t+3\).
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Substitute \(t=4\): \(S^{\prime}(4)=4+3=7\).
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Attach units: inches of depth per hour of time.
At 4 a.m. the snow is deepening at 7 inches per hour.
Common mistake
Watch out
With the same model, a student computes \(\frac{S(4)-S(0)}{4-0}=\frac{20}{4}=5\) and reports \(S^{\prime}(4)=5\).
Why it is wrong
That 5 is an average rate over four hours, found with two points. The derivative is an instantaneous rate, found from the rate formula at the single moment \(t=4\), which gives 7. Because the snow piles up faster as the night goes on, the rate at hour 4 is larger than the four-hour average.
Related terms and lessons
Browse the glossary: A–Z list · AP Calculus AB vocabulary · Calculus