t3_apply_derivatives
====================

.. py:function:: t3toolbox.backend.sampling_derivatives.t3_apply_derivatives(ww, pp, x, order)

   .. code-block:: python

      def t3_apply_derivatives(
              ww:     typ.Sequence[NDArray],  # probe vectors X,        len=d, elm_shape=W+(Ni,)
              pp:     typ.Sequence[NDArray],  # perturbation vectors P, len=d, elm_shape=W+(Ni,)
              x:      typ.Tuple[
                  typ.Sequence[NDArray],      # tucker_cores. len=d, elm_shape=C+(nUi,Ni)
                  typ.Sequence[NDArray],      # tt_cores.     len=d, elm_shape=C+(rLi,nUi,rR(i+1))
              ],                              # = TuckerTensorTrain.data
              order:  int,                    # highest derivative order
      ) -> NDArray:                           # apply-derivative jets, shape=(order+1,)+W+C


   Symmetric derivatives of applying a Tucker tensor train in all modes, in one repeated direction.

   The all-modes Euclidean analog of :py:func:`t3_probe_derivatives`: returns the stack
   ``y^(t) = d^t/ds^t apply(X, W + s P)|_0`` for ``t=0..order``, where ``apply`` contracts the tensor
   with the vectors in every mode (a scalar). Index ``0`` is the ordinary apply. Computed as the
   terminal carry of the left mu-jet sweep (via the cores), bond summed -- no right/central sweeps.
   Stacks ride as ``order + W + C`` (sample stack ``W``, frame stack ``C``; no tangent stack here).
   Verified against :py:func:`dense_apply_derivatives`.