Definition: The derivative of a function at a point is the instantaneous rate of change there, equal to the slope of the tangent line to its graph.
What is a derivative?
It is the limit of average rates of change: f'(x) = limh→0 (f(x + h) – f(x))/h. Geometrically it is the slope of the tangent line; physically, it is a velocity if f gives position.
How do you use it?
The power rule says the derivative of xn is n xn-1.
Worked example: f(x) = x3 – 4x. Then f'(x) = 3x2 – 4. At x = 3, f(3) = 27 – 12 = 15 and f'(3) = 27 – 4 = 23. The tangent line is y = 15 + 23(x – 3).
A positive derivative means the function is increasing; a negative derivative means it is decreasing.
What mistakes are common?
Differentiating a constant and getting the constant back; the derivative of a constant is 0. Another is using the average rate of change when a problem asks for the instantaneous one. Mathbot also writes the units: a derivative has units of output per input.