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The total derivative (ou derivative in the au sense of Fréchet) exists and equals the gradient of f of a (
) if
.
![{\displaystyle f(\mathbf {a+h} )=f(\mathbf {a} )+\langle \mathbf {\nabla f(\mathbf {a} )} ,\mathbf {h} \rangle +r(\mathbf {h} )}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/75b7b72123c6c7f4478a0dd8ba6cea4507958fd1)
As example R2,
, where h is a very small number, equals the sum:
![{\displaystyle f(a)}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/368cb4b81ba5754d7a354a4ce49c2f1084bdaace)
where
(straight line with gradient of the function at the point
)
- the rest
which depends only of h
So
[1]
See also[edit]
- ^ Analyse pour ingénieurs - semestre 2, C.A.Stuart, Lausanne, p. 25