Linearization of a non-linear function

$f(x) = \log(e^{x_1} + e^{x_2})$

admits the gradient at the point $x^0$ given by

$\nabla f(x_0) = \frac{1}{e^{x_1^0}+e^{x_2^0}}\left(\begin{array}{c} e^{x_1^0} \\ e^{x_2^0} \end{array}\right).$

Hence $f$ can be approximated near $x_0$ by the linear function

$f(x) \approx \log(e^{x_1} + e^{x_2})+ \frac{1}{e^{x_1^0}+e^{x_2^0}}\left((x_1 - x_1^0)e^{x_1^0} + (x_2 - x_2^0)e^{x_2^0}\right)$

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