A straight line has a slope because two points tell us how much the line rises for how much it runs.
m = Δy / Δx
But a curve changes direction from place to place.
How can we talk about the slope at just one point when slope seems to require two points?
That question is one of the doors into calculus.
Quick Answer
Take a curve y=f(x) and a fixed point P=(a,f(a)).
Choose a nearby point Q=(a+h,f(a+h)).
The secant slope is:
msecant = [f(a+h)−f(a)] / h
Now let h→0. If the secant slopes approach one value, that limiting value is the tangent slope — the derivative.[1]
f′(a) = limh→0 [f(a+h)−f(a)] / h
Two Points Give a Secant Line
Use the curve y=x².
Choose P=(1,1) and Q=(2,4).
The line through them is a secant line.
m = (4−1)/(2−1) = 3
This slope describes the curve between two points, not exactly at P.
Figure 1. Two points give a secant line and an average slope between them.
Move the Second Point Closer
Keep P fixed at (1,1).
Move Q from x=2 to x=1.5: the slope becomes 2.5.
Move Q to x=1.1: the slope becomes 2.1.
3 → 2.5 → 2.1 → ...
Figure 2. As Q moves toward P, the secant lines approach a limiting direction.
Use h to Describe the Gap
Let the horizontal separation be h.
The nearby point is Q=(1+h,(1+h)²).
The secant slope is:
[(1+h)²−1]/h
Since (1+h)² = 1+2h+h²:
[(1+h)²−1]/h = 2+h
| h | Secant slope |
|---|---|
| 1 | 3 |
| 0.5 | 2.5 |
| 0.1 | 2.1 |
| 0.01 | 2.01 |
| −0.01 | 1.99 |
| −0.1 | 1.9 |
| −0.5 | 1.5 |
Figure 3. The secant slopes approach 2 from both sides as h approaches 0.
Why Not Just Set h = 0?
If we set h=0 inside the secant formula, we divide by zero.
So we ask what the slope values approach as h gets arbitrarily close to zero.
limh→0(2+h)=2
This reuses a distinction we met much earlier: approaching something is not the same as being equal to it.
Now We Can Name It: The Derivative
f′(a) = limh→0 [f(a+h)−f(a)]/h
The derivative is the limiting value of nearby secant slopes.[2]
Tangent Does Not Simply Mean “Touches Once”
A tangent line may cross the curve at the point of tangency or meet the curve again elsewhere.
The tangent line gives the limiting local direction of the curve.
Zoom In: The Curve Becomes Locally Almost Straight
From far away, the parabola is clearly curved. Near P=(1,1), the curve and tangent line become increasingly hard to distinguish.
Figure 4. Near a smooth point, the curve becomes locally almost indistinguishable from its tangent.
This is local linearity.
The Tangent Line at P=(1,1)
We found f′(1)=2.
The tangent line through (1,1) with slope 2 is:
y−1 = 2(x−1)
or:
y = 2x−1
Try It — Find the Slope at x=2
For y=x²:
[(2+h)²−4]/h = 4+h
As h→0, 4+h→4.
Therefore f′(2)=4.
What If the Slopes Do Not Settle?
A derivative does not have to exist everywhere. Sharp corners, vertical tangents, and other irregular behavior can prevent a unique ordinary finite tangent slope from existing.
From Slope to Instantaneous Velocity
On a position-time graph, average velocity is a secant slope:
vavg = Δx / Δt
As Δt→0, the limiting tangent slope is instantaneous velocity.[3]
Figure 5. Average velocity is a secant slope; instantaneous velocity is the limiting tangent slope.
Words to Keep
Secant line
A line through two points on a curve.
Secant slope
The slope between two points on a curve.
Tangent line
The limiting local line approached by nearby secants, when that limit exists.
Difference quotient
An expression such as [f(a+h)−f(a)]/h that gives a secant slope.
Limit
A precise description of the value a quantity approaches.
Derivative
The limiting rate of change; geometrically, the tangent slope at a point when the derivative exists.
Local linearity
The idea that a smooth curve looks increasingly like its tangent line when viewed sufficiently close to a point.
One Sentence to Keep
The slope at one point is the value approached by slopes between two points as the second point moves toward the first.
What Should We Ask Next?
We now know how to define the slope of a curve at one point.
On a position-time graph, that slope tells us how fast position is changing.
Next: How Can a Graph Tell You How Fast Something Is Moving Right Now?
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