Derivative

Derivative
Tag	`0F1WR9C83K`
URI	`m2://0F1WR9C83K`
Kind	definition
Writing	W1 stated
MSC	26A24
Defines	Derivative
Uses	Limit of a function

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Definition

Let $I$ be an open interval, let $f \colon I \to \mathbb{R}$ be a function, and let $a \in I$ . A real number $L$ is the derivative of $f$ at $a$ if $\lim_{h \to 0} \frac{f(a + h) - f(a)}{h} = L.$ The notation is $f'(a) = L$ .

Remarks

The quotient is the slope of the line through $(a, f(a))$ and $(a + h, f(a + h))$ . The limit is a limit of a function of $h$ at $0$ . The derivative is unique when it exists. For example, the derivative of $x^2$ at $a$ is $\lim_{h \to 0} (2a + h) = 2a$ . Mathlib defines the relation " $f$ has derivative $L$ at $a$ " as HasDerivAt.

Explanations

L1 human

The derivative of $f$ at $a$ is the instantaneous rate of change of $f$ at $a$ . Geometrically, it gives the steepness of the graph at the point $(a, f(a))$ .

What this node needs for W2

edge instance_of in: one example (0 of 1)
edge not_instance_of in: one non-example that fails one clause (0 of 1)

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What this needs

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Limit point of a set definition
Limit of a function definition
Derivative definition

What depends on this

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

Power series are differentiable inside the radius of convergence 002X7EK91H
A function with zero derivative is constant 003QDZ4T16
Natural logarithm 0044CPHJBV
A differentiable function is convex iff its derivative is increasing 005JT76HCK
Taylor's theorem with the Cauchy remainder 005Q84BMAD
Euler–Mascheroni constant 00C5P9JZBK
The logarithm grows slower than any power 00DGYY10FV
Derivative of sine 00R4MWW532
1 + x is at most exp x 012C7GFF20
Reduction formula 012RYET5E2
Rolle's theorem 0194DY35BG
Weierstrass function 01A0MK1730
First derivative test 01A2EGXJEM
cos pi = -1 01A8JGR4JM
sin x / x tends to 1 01QSSKS5C1
Tangent line 01R1HW3Y42
Taylor series 0225QPWRRE
A function with nonnegative derivative is increasing 026BJBQNGJ
Derivative function 027K9RMZ43
Angle addition formulas 028XEQCE88
Inverse trigonometric functions 02HTDK1FBD
Power series of sine 02KTRHQMHN
Cauchy mean value theorem 02SYZAWM2K
Leibniz formula for pi 037FD5NBRT
Integral of exp(-x) over the half line 039JAF7ZAN
Taylor's theorem with the Peano remainder 03AJHX431A
Pythagorean trigonometric identity 03H7WZ4T3K
Taylor's theorem 03QHBVW4TT
Indefinite integral 03VHANA6ZK
Maclaurin series 043SJ5RF59
log x is at most x - 1 04CBRA4JRK
Inverse hyperbolic functions 04D7CHT5V1
Integral of cosine 04WTBQ7APQ
Derivative of arctangent 05P4B4Q22V
Harmonic numbers grow like log n 05S2GSPMGA
First fundamental theorem of calculus 063H5KJC69
Arctangent 065XW2V6C3
Integral of the exponential function 06D0EWR2DV
Power series of the exponential function 06JZGMYTYD
Continuously differentiable function 06SKCHDQT5
Tangent function 06T0HS14PP
Newton's method 07EXVNF9VP
A function with positive derivative is strictly increasing 07J5X2T1ZX
Derivatives of the hyperbolic functions 07T2TKGZ4J
Derivative of the logarithm 07X1QA6BRW
Differentiable functions are continuous 07ZMX4ANJS
Higher derivative 084NGSTWM7
Power series of cosine 08W5RN375X
Integration by parts 08Y1BBQTK9
Smooth non-analytic function 09RTYYHXKA
Differentiable function 0A61B60B5R
Arcsine 0B89D93YA1
Mean value theorem 0BARQY52N5
Smooth function 0BE3QDH73A
Quotient rule 0BS0NJPKVE
Antiderivative 0BW9N5CDSB
pi 0C1VYVNV2N
Absolute value is continuous but not differentiable at 0 0CQ9ZM383N
Uniform convergence of derivatives allows term-by-term differentiation 0CS42KEM5C
Integration by substitution 0D37QFPEZE
The exponential function is convex 0D7GRZ2ADX
Basel problem 0DMY75D6FV
Trigonometric substitution 0EAR3KNRTN
The exponential function is its own derivative 0EX9GFJ0B3
The gamma function extends the factorial 0FDCW77H44
Second derivative test 0FZFHK97P7
Derivative of arcsine 0GA6G43EA9
Critical point 0GEQ42PTGB
Area under a curve 0GXA8Z205E
Integral of sine 0GXCXCX3QN
Dirichlet integral 0GZNSHMG4Q
Taylor's theorem with the Lagrange remainder 0H5ZAZCX2T
Taylor polynomial 0HFSFVMWQB
One-sided derivative 0HXEX0DPRA
Logarithm of a quotient 0J7PHRBNNS
L'Hôpital's rule 0JWT4PPQ7B
Power rule 0JZGW75C5Z
Integral of 1/x 0K1N1Y4EGZ
Logarithm of a product 0K2FNY6DYV
General Leibniz rule 0K9QCK6WJC
Stirling's formula 0KC0F51T6A
The exponential function is strictly increasing 0KN2V7840E
Derivative of x^a 0KY08TWDMQ
Derivative of an inverse function 0M0RZ3HSB0
Linearity of differentiation 0M0TKFRX2A
Derivative of a polynomial 0MH1WK89PP
The integral of a continuous function is an antiderivative 0NC2A3HWGT
Fundamental theorem of calculus 0NHGY0N0JB
Sine is 2π-periodic 0P4NQD746K
A function with bounded derivative is Lipschitz 0P64VE79X3
Fermat's theorem on stationary points 0PCVBRV9VJ
Integral of a power 0Q3HYFT9RQ
Darboux's theorem 0Q5A47J48D
Reciprocal rule 0QD4E5JYPT
Irrationality of pi 0QHZX1PK8V
pi is greater than 3 0R7SVNYTJS
Arccosine 0RDP2HEFEW
Logarithm is the inverse of the exponential 0RKW2X63YR
A continuous nowhere differentiable function 0RXK2Z808Q
Chain rule 0RXX2PT1MQ
Power-reduction formula for cosine 0S049HG1W5
Indeterminate form 0S7G92KKA7
Derivative of tangent 0SAMR34WXC
Power series of arctangent 0T32ME0A6H
Term-by-term differentiation 0TJ8QBJX5K
Real exponentiation 0TV07AQT62
Derivative of cosine 0V2DV29EGT
Logarithm of a power 0V948DE5KA
Product rule 0V9A43H7WV
Improper integral of x^p on an unbounded interval 0VGKB2NC4T
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 fundamental theorem of calculus 0XYAFCN0PT
Second derivative 0Y9D2FJRQ4
Mercator series 0YZXNE3HFR
A function with nonnegative second derivative is convex 0Z41C17TGP
Constant multiple rule 0Z9JB71BS0
The logarithm is strictly concave 0ZS54B9GJF

Sources

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

What links here

Used by

sin x / x tends to 1 01QSSKS5C1
Tangent line 01R1HW3Y42
Derivative function 027K9RMZ43
First fundamental theorem of calculus 063H5KJC69
Newton's method 07EXVNF9VP
Differentiable function 0A61B60B5R
Antiderivative 0BW9N5CDSB
The exponential function is its own derivative 0EX9GFJ0B3
Critical point 0GEQ42PTGB
One-sided derivative 0HXEX0DPRA
Power rule 0JZGW75C5Z
Derivative of an inverse function 0M0RZ3HSB0
Linearity of differentiation 0M0TKFRX2A
Fermat's theorem on stationary points 0PCVBRV9VJ
Reciprocal rule 0QD4E5JYPT
Chain rule 0RXX2PT1MQ
Product rule 0V9A43H7WV