Derivative
The derivative of f at a is the limit of (f(a + h) - f(a)) / h as h tends to 0. It measures the instantaneous rate of change of f at a.
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Analogy
A speedometer: it shows your speed right now, not your average over the trip.
Example
(f(3+h) - f(3))/h = (9 + 6h + h^2 - 9)/h.This simplifies to 6 + h.As h tends to 0, it tends to 6.So f'(3) = 6.
In practice
Derivatives tell you how steep a curve is, how fast a quantity grows, and where it peaks.
Practice
Flashcards
- Give the limit definition of f'(a).The limit as h tends to 0 of (f(a + h) - f(a)) / h.
Interview questions
- Find the equation of the tangent line to y = x^2 at x = 3, and explain what the derivative represents.
- f(3) = 9 gives the point (3, 9)
- f'(x) = 2x so the slope is 6
- Tangent line y - 9 = 6(x - 3), so y = 6x - 9
- Derivative is the instantaneous rate of change
Questions people ask
What does the derivative actually measure?
It measures how quickly the output changes when the input changes by a tiny amount. Graphically, it is the slope of the tangent line at a point. For a car's position, the derivative is its speed at that exact moment.
Why is h in the definition of the derivative?
The number h is a small step from a to a + h. The fraction is the slope of a line through two nearby points. As h shrinks to zero, that line turns into the tangent line, and its slope becomes the derivative.