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m The description is now slightly more precise. We don't take the derivation of "S" but the derivation of it's approximation we just calculated. |
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&= \left [\mathbf y - \mathbf f\left (\boldsymbol\beta\right )\right ]^{\mathrm T}\left [\mathbf y - \mathbf f\left (\boldsymbol\beta\right )\right ] - 2\left [\mathbf y - \mathbf f\left (\boldsymbol\beta\right )\right ]^{\mathrm T} \mathbf J \boldsymbol\delta + \boldsymbol\delta^{\mathrm T} \mathbf J^{\mathrm T} \mathbf J\boldsymbol\delta.
\end{align}</math>
Taking the derivative of this approximation of <math>S\left (\boldsymbol\beta + \boldsymbol\delta\right )</math> with respect to {{tmath|\boldsymbol\delta}} and setting the result to zero gives
:<math>\left (\mathbf J^{\mathrm T} \mathbf J\right )\boldsymbol\delta = \mathbf J^{\mathrm T}\left [\mathbf y - \mathbf f\left (\boldsymbol\beta\right )\right ],</math>
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