Math 2.0

Real functions Definition

Derivative

Builds on Limit of a function

Definition

Let II be an open interval, let f ⁣:I→ℝf \colon I \to \mathbb{R} be a function, and let a∈Ia \in I. A real number LL is the derivative of ff at aa if lim⁡h→0f(a+h)−f(a)h=L.\lim_{h \to 0} \frac{f(a + h) - f(a)}{h} = L. The notation is f′(a)=Lf'(a) = L.

Remarks

The quotient is the slope of the line through (a,f(a))(a, f(a)) and (a+h,f(a+h))(a + h, f(a + h)). The limit is a limit of a function of hh at 00. The derivative is unique when it exists. For example, the derivative of x2x^2 at aa is lim⁡h→0(2a+h)=2a\lim_{h \to 0} (2a + h) = 2a. Mathlib defines the relation "ff has derivative LL at aa" as HasDerivAt.

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Limit point of a set, definition, no evidence levelLimit point of a setdefinitionLimit of a function, definition, no evidence levelLimit of a functiondefinitionDerivative, definition, no evidence levelDerivativedefinition
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  1. Limit point of a set definition
  2. Limit of a function definition
  3. Derivative definition

Where this leads

214 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Derivative, As a limit
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