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Differentiation
Differentiability at a point
What it means to have a tangent
A function can be continuous without being differentiable: continuity rules out jumps, differentiability rules out corners. It is the difference between drawing without lifting the pen and drawing without a kink.
The difference quotient
📖 Definition
f is differentiable at a if the difference quotient between a and a + h has a finite limit as h tends to 0. That limit is the derivative f′(a).
f′(a) = limh→0 f(a + h) − f(a)h
The slope of the secant tends to the slope of the tangent.
✏️ Example
x ↦ |x| is not differentiable at 0: the quotient is −1 on the left and +1 on the right. The two limits differ, so there is no tangent.
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Calculus · Differentiation
2 / 7
lessons completed
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7 cards due today
Lessons
The concept of a limit
Completed · 92%
Limits and operations
Completed · 86%
In progress · section 3 of 5
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Rolle’s theorem
Up next · 9 min
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Final exam
5 lessons left to complete
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Explain Rolle’s theorem to me
If f is continuous on [a, b], differentiable on (a, b) and f(a) = f(b), then there is a c in (a, b) with f′(c) = 0: the tangent is horizontal there.
Example: f(x) = x² − 4x satisfies f(1) = f(3) = −3, and indeed f′(x) = 2x − 4 vanishes at c = 2.
What does Rolle’s theorem guarantee?
Correct. Rolle gives you that c exists, not its value.
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