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

Because of the equation
*S*^{2} = *S*^{2}_{f}⋅*S*^{2}_{s} and the form of the extended spectral density function (15.63) a convolution of the model-free space occurs if the model-free parameters {*S*^{2}_{f}, *S*^{2}_{s}, *τ*_{f}, *τ*_{s}} are optimised rather than the parameters {*S*^{2}, *S*^{2}_{f}, *τ*_{f}, *τ*_{s}}. This convolution increases the complexity of the gradient. For completeness the first partial derivatives are presented below.

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} + + . |
(15.91) |

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

= c_{i}τ_{i} - + . |
(15.92) |

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

= S^{2}_{f}c_{i}τ_{i} - . |
(15.93) |

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

= (1 - S^{2}_{f})c_{i}τ_{i}^{2}. |
(15.94) |

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

= S^{2}_{f}(1 - S^{2}_{s})c_{i}τ_{i}^{2}. |
(15.95) |

The relax user manual (PDF), created 2016-10-28.