docstr: Derivative of the degree-2 polynomial fit of the given x, y pairs
  
  Directly computes the derivative of the least-squares-fit quadratic
  function estimated from (x[...],y[...]) pairs.
  
  Parameters
  ----------
  x : array, shape (n,)
      increasing array representing the x-coordinates of the points to
      be fit
  y : array, shape (n,)
      array representing the y-coordinates of the points to be fit
  
  Returns
  -------
  y0 : float
      the estimated derivative at x0 = 0.5*len(x)
code: def _approximate_derivative_direct(self, x, y):
      """Derivative of the degree-2 polynomial fit of the given x, y pairs
  
      Directly computes the derivative of the least-squares-fit quadratic
      function estimated from (x[...],y[...]) pairs.
  
      Parameters
      ----------
      x : array, shape (n,)
          increasing array representing the x-coordinates of the points to
          be fit
      y : array, shape (n,)
          array representing the y-coordinates of the points to be fit
  
      Returns
      -------
      y0 : float
          the estimated derivative at x0 = 0.5*len(x)
      """
      x = np.asarray(x)
      y = np.asarray(y)
      X = np.row_stack((x**2, x, np.ones_like(x)))
      XX = X.dot(X.T)
      b = X.dot(y)
      beta = npl.solve(XX, b)
      x0 = 0.5 * len(x)
      y0 = 2.0 * beta[0] * x0 + beta[1]
      return y0
  
imports: import logging
  import abc
  import numpy
  import numpy.linalg
  import nibabel
  from nibabel.streamlines import ArraySequence
  from dipy.align import vector_fields
  from dipy.align import floating
  from dipy.align import VerbosityLevels
  from dipy.align import Bunch
  from dipy.align.scalespace import ScaleSpace
label_desc: Use image node_id 10 for calling the SymmetricDiffeomorphicRegistration obj's underlying member method code with example usage: obj._approximate_derivative_direct(x, y) and returns: y0