Higher derivative

Higher derivative
Tag	`084NGSTWM7`
URI	`m2://084NGSTWM7`
Kind	definition
Writing	W0 stub
MSC	26A24
Defines	Higher derivative
Uses	Derivative function

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Definition

Let $I$ be an open interval and let $f \colon I \to \mathbb{R}$ . Let $f^{(0)} = f$ . If $f^{(n)}$ is defined and differentiable on $I$ , let $f^{(n+1)} = (f^{(n)})'$ , its derivative function. The function $f^{(n)}$ is the $n$ -th derivative of $f$ , and $f$ is $n$ times differentiable when it exists.

Remarks

The second derivative is $f'' = f^{(2)}$ . For example, the $n$ -th derivative of $x^n$ is the constant $n!$ . Mathlib defines iteratedDeriv n f.

Explanations

L1 human

Higher derivatives come from differentiating again and again. They describe finer features of a function, like the bending of its graph.

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Limit point of a set definition
Limit of a function definition
Derivative definition
Differentiable function definition
Derivative function definition
Higher derivative definition

What depends on this

57 nodes use this node, directly or through other nodes.

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cos pi = -1 01A8JGR4JM
sin x / x tends to 1 01QSSKS5C1
Taylor series 0225QPWRRE
Angle addition formulas 028XEQCE88
Inverse trigonometric functions 02HTDK1FBD
Power series of sine 02KTRHQMHN
Leibniz formula for pi 037FD5NBRT
Taylor's theorem with the Peano remainder 03AJHX431A
Pythagorean trigonometric identity 03H7WZ4T3K
Taylor's theorem 03QHBVW4TT
Maclaurin series 043SJ5RF59
log x is at most x - 1 04CBRA4JRK
Integral of cosine 04WTBQ7APQ
Derivative of arctangent 05P4B4Q22V
Arctangent 065XW2V6C3
Power series of the exponential function 06JZGMYTYD
Continuously differentiable function 06SKCHDQT5
Tangent function 06T0HS14PP
Power series of cosine 08W5RN375X
Integration by parts 08Y1BBQTK9
Smooth non-analytic function 09RTYYHXKA
Arcsine 0B89D93YA1
Smooth function 0BE3QDH73A
pi 0C1VYVNV2N
Integration by substitution 0D37QFPEZE
The exponential function is convex 0D7GRZ2ADX
Basel problem 0DMY75D6FV
Trigonometric substitution 0EAR3KNRTN
The gamma function extends the factorial 0FDCW77H44
Second derivative test 0FZFHK97P7
Derivative of arcsine 0GA6G43EA9
Integral of sine 0GXCXCX3QN
Dirichlet integral 0GZNSHMG4Q
Taylor's theorem with the Lagrange remainder 0H5ZAZCX2T
Taylor polynomial 0HFSFVMWQB
General Leibniz rule 0K9QCK6WJC
Stirling's formula 0KC0F51T6A
Sine is 2π-periodic 0P4NQD746K
Irrationality of pi 0QHZX1PK8V
pi is greater than 3 0R7SVNYTJS
Arccosine 0RDP2HEFEW
Power-reduction formula for cosine 0S049HG1W5
Derivative of tangent 0SAMR34WXC
Power series of arctangent 0T32ME0A6H
Derivative of cosine 0V2DV29EGT
Tangent half-angle substitution 0W8ETY2AZ6
Sine and cosine 0WB57D9488
Integration by partial fractions 0WBTWFXJQG
x^2 sin(1/x) is differentiable with discontinuous derivative 0WDAKKN4NC
Taylor's theorem with the integral remainder 0WJGR81HBD
Wallis product 0XCYA4YERS
Second derivative 0Y9D2FJRQ4
A function with nonnegative second derivative is convex 0Z41C17TGP