The Derivative at a Point
The slope of the
tangent line at a point
has a name: the derivative of f at that point.
We write it f'(a), read "f prime of a".
It is defined as the limit of the difference quotient as the step shrinks to zero:
f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h}
In words: f'(a) is the instantaneous rate of
change of f at x = a — how
fast the output is changing at that exact instant.
Computing one: f(x) = x^2 at a = 3
Let us find f'(3). Build the difference quotient, simplify, then
let the limit do its work. Start with
f(3 + h) = (3 + h)^2 = 9 + 6h + h^2 and
f(3) = 9:
\frac{(9 + 6h + h^2) - 9}{h} = \frac{6h + h^2}{h} = \frac{h(6 + h)}{h} = 6 + h
Now the denominator is gone, so we can find the
limit by
substitution — just put h = 0 into the simplified
expression:
f'(3) = \lim_{h \to 0} (6 + h) = 6
So at x = 3 the curve y = x^2 has slope
6. The tangent line there rises six units for every one across.
See the tangent at any point
Slide the point a along f(x) = x^2. The
bold tangent line always has slope f'(a) = 2a — flat at the
bottom (a = 0), steeper as you move out, and negative on the left.
The recipe, step by step
Every derivative-at-a-point follows the same four moves. Step through them for a fresh
example.
Watch it explained
Sal Khan calculates the slope of a tangent line at a point using the definition of the
derivative.