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derivative (Definition)

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"derivative" is owned by rmilson. [ full author list (8) | owner history (1) ]
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See Also: table of derivatives, derivative of inverse function, partial derivative, gradient, related rates, Lipschitz condition and differentiability

Also defines:  directional derivative, Fréchet derivative

Attachments:
higher order derivatives of sine and cosine (Derivation) by pahio
tangent line (Definition) by Mathprof
one-sided derivatives (Definition) by pahio
derivative of $x^n$ (Theorem) by Algeboy
alternative proof of derivative of $x^n$ (Proof) by Wkbj79
derivatives by pure algebra (Definition) by Algeboy
Fréchet derivative is unique (Theorem) by Mathprof
higher order derivatives (Definition) by PrimeFan
logarithmic derivative (Definition) by rspuzio
derivative for parametric form (Derivation) by pahio
table of derivatives (Feature) by CWoo
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Cross-references: variations, components, measure, vector fields, operator, open set, distributions, pushforward, connection, well defined, admissible, domain, map, tangent spaces, isomorphisms, canonical, neighborhoods, charts, related rates, structure, locally homeomorphic, topological space, manifolds, finite dimensional, wave equations, applications, variables, independent, partial derivatives, extension, operator norm, norm, bilinear, properties, Jacobian, multiplication, distributive properties, linear operator, vector, bounded linear map, scalar, Banach spaces, areas, similar, near, chain rule, product rule, Calculus, quotient rule, expressions, derivative notation, elementary functions, limit, real function, vicinity, approximation, slope, tangent line, smooth, curves, ratio, equation, line, straight, point, function
There are 219 references to this entry.

This is version 31 of derivative, born on 2002-05-31, modified 2009-04-25.
Object id is 2975, canonical name is Derivative2.
Accessed 53791 times total.

Classification:
AMS MSC26A24 (Real functions :: Functions of one variable :: Differentiation : general theory, generalized derivatives, mean-value theorems)
 46G05 (Functional analysis :: Measures, integration, derivative, holomorphy :: Derivatives)
 26B05 (Real functions :: Functions of several variables :: Continuity and differentiation questions)

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Discussion
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Additional Reference by smithpith on 2009-04-25 22:53:30
PlanetMath article: Non-Newtonian calculus.
[ reply | up ]
Derivative as Linear Approximation by joshsamani on 2008-04-13 05:23:40
I understand that the Frechet derivative is a linear approximation in the sense that it is a linear transformation which approximates a function between Banach spaces.

But, I have read in multiple texts that it is the "best" linear approximation. In what sense in the Frechet derivative the "best" linear approximation as opposed to just some linear approximation?
[ reply | up ]
Possible error in formula by dtowell on 2003-12-09 12:13:10
Under Linearity, it seems like there is a typo in the formula, bg(x) becomes bf'(x) which doesn't seem right but its been a long time since I studied this sort of thing so I could be wrong.

Dwayne

[ reply | up ]
Norm on L(V,W) by igor on 2002-05-31 17:55:35
I mention in the article above that L(V,W)
can be considered a Banach space itself.


The fact that it is a vector space is trivial.
However, I'm not sure what the canonical norm
for that space is. Perhaps the L^p norm?
What are some other commonly used norms that
turn L(V,W) into a normed space? Do they
generate the same topology as the L^p norm?
[ reply | up ]

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