The definition of the derivative
Slope at a point
A line has the same slope everywhere. A curve does not. A slope needs two points, so what could the slope of a curve at a single point mean?
Take the point P=(a,f(a)) and a second point Q=(a+h,\,f(a+h)) on the curve. The ochre secant line through P and Q has a slope you can compute. Drag Q toward P. What happens to that slope as h shrinks?
Drag Q along the curve to change h, or drag P to move the point. Turn on "Zoom in as h shrinks" to see the curve straighten out near P.
| |h| | h > 0 | h < 0 |
|---|
Reading the definition
The derivative of f at a is
provided this limit exists. Each part matches something in the graph above.
- f(a+h)-f(a)
- The rise: how much the output changes between P and Q.
- h
- The run: how far Q is from P horizontally. It may be positive or negative, but it is never 0, since dividing by 0 is undefined.
- The quotient
- The slope of the secant line through P and Q. This is an ordinary slope between two points.
- \lim_{h\to 0}
- The value the secant slopes approach as Q slides toward P from either side. If the slopes from the left and from the right approach the same number, that number is the slope of the tangent line.
Writing x=a+h gives an equivalent form that some texts prefer:
Because f'(a) is a limit, it carries the ε-δ meaning: for every \varepsilon>0 there is a \delta>0 such that 0<|h|<\delta puts the difference quotient within \varepsilon of f'(a).
Computing a derivative from the definition
Setting h=0 in the difference quotient always gives \tfrac{0}{0}, which tells you nothing. The work is to simplify the quotient until the factor of h in the denominator cancels. Try each step yourself before revealing it.
Solve it step by step
Find each derivative from the limit definition, then check it against the graph. The score counts steps you answer correctly on the first try.
When there is no derivative
The limit has to exist from both sides and agree. For f(x)=|x| at a=0, the difference quotient is \tfrac{|h|}{h}. What is it when h>0? When h<0? The two one-sided limits are 1 and -1, so |x| has no derivative at 0. The graph has a corner there.
For f(x)=\sqrt{x} at a=0, only h>0 makes sense, and the quotient is \tfrac{\sqrt h}{h}=\tfrac{1}{\sqrt h}. It grows without bound as h\to 0^+. The secant lines approach a vertical line, which has no slope.
Match the tangent
Five rounds. Each round marks a point on a curve. Tilt the ochre line until you think it is tangent at that point, then check. An answer within 2° of the true tangent earns 2 points, and one within 6° earns 1 point.