Slope needs two points. A curve has a different steepness at every point, so which two?
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Start with two
Take a point and a nearby one. The line through them is a secant, and its slope is an average.
Then move them together
As the second point slides towards the first, the secant tilts towards the tangent.
The definition
f′(x) is the limit of (f(x + h) − f(x)) ÷ h as h approaches 0. It is a slope, defined by a limit.
The problem the derivative solves
Average speed over an interval is easy. Speed at an instant has no interval, so the usual ratio gives 0/0. The derivative is the limit that resolves this, and the construction is worth following once in full.
The difference quotient
(f(a + h) − f(a))/h is the slope of the secant over an interval of width h. Letting h approach zero slides the second point onto the first, and the limit of those slopes is the tangent slope.
Notation
f′(a), dy/dx and df/dx all denote the derivative. Leibniz's dy/dx form is the most suggestive — it looks like a ratio and behaves like one in the chain rule — which is why it survived.
The derivative has many readings
Slope of the tangent, instantaneous rate of change, velocity, marginal cost, or sensitivity to a parameter. All are the same limit; which reading matters depends on what the function represents.
Step 2: Try It Yourself
Slide the second point onto the first and watch the secant become the tangent.
- Point(2, 2)
- Second point(5, 23)
- Slope between them7
Slide the second point towards the first. The slope between them approaches the slope of the curve at that point.
Step 3: Watch an Example
One step at a time.
Watch Priya Differentiate x² by Hand
Priya finds the derivative of f(x) = x² from the definition.
- Step 1
The difference quotient is ((x + h)² − x²) ÷ h.
Step 4: Your Turn
Practice makes it stick.
The Slope
Problem 1 of 2
f(x) = x², so f′(x) = 2x. What is the slope at x = 3?
The Turning Point
Problem 2 of 2
f(x) = x². At what x is the slope zero?
Slopes on a Curve
1 of 4
f′(x) = 2x. Slope at x = 5?
2 of 4
f′(x) = 2x. Slope at x = −4?
3 of 4
f(x) = 3x. What is f′(x)?
4 of 4
What does the secant become as the points meet?
Step 5: Quick Check
Show what you know.
Question 1 of 2
f′(x) = 2x. Slope at x = 7?
Question 2 of 2
What is a derivative?
What You Learned
- A secant slope is an average over an interval.
- As the interval shrinks to nothing, the secant becomes the tangent.
- The derivative is that limit, and it is a slope.