Math 2.0

definition

Derivative function

Definition

Let II be an open interval and let f ⁣:I→Rf \colon I \to \mathbb{R} be differentiable on II. The derivative function of ff is the function f′ ⁣:I→Rf' \colon I \to \mathbb{R} that maps each x∈Ix \in I to the derivative f′(x)f'(x) of ff at xx.

Remarks

Other notations are dfdx\frac{df}{dx} and DfD f. For example, the derivative function of x3x^3 is 3x23 x^2. The derivative function need not be continuous. See x^2 sin(1/x). Mathlib defines deriv f on all of R\mathbb{R}, with value 00 at points where ff is not differentiable.

Explanations

L1 human

The derivative function collects the slopes of ff at all points into one new function. We can then differentiate it again.

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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 levelDerivativedefinitionDifferentiable function, definition, no evidence levelDifferentiable functi…definitionDerivative function, definition, no evidence levelDerivative functiondefinition
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  1. Limit point of a set definition
  2. Limit of a function definition
  3. Derivative definition
  4. Differentiable function definition
  5. Derivative function definition

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Sources

  1. Web https://en.wikipedia.org/wiki/Derivative, Definition

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