Fungrim entry: 452407

Symbol: RealDerivative —

\frac{d}{d x}\, f\!\left(x\right)

— Real derivative

The call syntax for this operator is the same as for Derivative.

The result is defined as

f'(x) = \lim_{h \to 0} \frac{f\!\left(x + h\right) - f\!\left(x\right)}{h}

where the limit is taken with respect to a real variable

h

( RealLimit ).

Note that

x

can be complex and that the "real derivative" can be complex-valued; the "real" qualifier just refers to the direction in which the limit is computed.

Definitions:

Fungrim symbol	Notation	Short description
`RealDerivative`	$\frac{d}{d x}\, f\!\left(x\right)$	Real derivative
`Derivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Derivative
`RealLimit`	$\lim_{x \to a} f\!\left(x\right)$	Limiting value, real variable

Source code for this entry:

Entry(ID("452407"),
    SymbolDefinition(RealDerivative, RealDerivative(Call(f, x), x, x), "Real derivative"),
    Description("The call syntax for this operator is the same as for", SourceForm(Derivative), "."),
    Description("The result is defined as", Equal(RealDerivative(f(x), x, x), RealLimit(Div(Sub(f(Add(x, h)), f(x)), h), h, 0)), "where the limit is taken with respect to a real variable", h, "(", SourceForm(RealLimit), ")."),
    Description("Note that", x, "can be complex and that the \"real derivative\" can be complex-valued; the \"real\" qualifier just refers to the direction in which the limit is computed."))

Topics using this entry

Operators