- partial derivative
- partial derivative
*S*^{2}partial derivative*S*^{2}_{f}partial derivative*τ*_{f}partial derivative*τ*_{s}partial derivative

The model-free gradient of the extended spectral density function (15.63) is the vector of partial derivatives of the function with respect to the geometric parameter
, the orientational parameter
, the order parameters *S*^{2} and *S*^{2}_{f}, and the internal correlation times *τ*_{f} and *τ*_{s}.
The positions in the vector correspond to the model parameters which are being optimised.

The partial derivative of (15.63) with respect to the geometric parameter is

The partial derivative of (15.63) with respect to the orientational parameter is

= τ_{i} + + . |
(15.72) |

The partial derivative of (15.63) with respect to the order parameter *S*^{2} is

= c_{i}τ_{i} - . |
(15.73) |

The partial derivative of (15.63) with respect to the order parameter *S*^{2}_{f} is

= - c_{i}τ_{i} - . |
(15.74) |

The partial derivative of (15.63) with respect to the correlation time *τ*_{f} is

= (1 - S^{2}_{f})c_{i}τ_{i}^{2}. |
(15.75) |

The partial derivative of (15.63) with respect to the correlation time *τ*_{s} is

= (S^{2}_{f} - S^{2})c_{i}τ_{i}^{2}. |
(15.76) |

The relax user manual (PDF), created 2020-08-26.