The derivative is the limit of the difference quotient; geometrically, the slope of the tangent line to the graph.
Symbols
f′(x0)
derivative at the point
x0
point
h
increment of the argument
Where it comes from
The derivative at a point is the limit of the slope of the secant line as the second point approaches the first — that is, the slope of the tangent line. It tells you how fast the function is growing at that point.
When to use it
Definition; the basis for computing tangent lines, extrema, and instantaneous velocity.
Example
What does f′(x0)=3 mean?
Substitutionf′(x0) = slope of the tangent line.
ResultAt the point x0 the function grows at a rate of 3 (the tangent line has slope 3).