Subsections

## The alternative extended model-free gradient

Because of the equation S2 = S2fS2s and the form of the extended spectral density function (15.63) a convolution of the model-free space occurs if the model-free parameters {S2f, S2s, τf, τs} are optimised rather than the parameters {S2, S2f, τf, τs}. This convolution increases the complexity of the gradient. For completeness the first partial derivatives are presented below.

### partial derivative

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

### partial derivative

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

 = τi + + . (15.91)

### S2f partial derivative

The partial derivative of (15.63) with respect to the order parameter S2f is

 = ciτi - + . (15.92)

### S2s partial derivative

The partial derivative of (15.63) with respect to the order parameter S2s is

 = S2fciτi - . (15.93)

### τf partial derivative

The partial derivative of (15.63) with respect to the correlation time τf is

 = (1 - S2f)ciτi2. (15.94)

### τs partial derivative

The partial derivative of (15.63) with respect to the correlation time τs is

 = S2f(1 - S2s)ciτi2. (15.95)

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