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

$nabla f(x_0) = frac{1}{e^{x^0_1}+e^{x^0_2}} left(begin{array}{c} e^{x_1^0} e^{x_2^0} end{array}right) .$

Hence can be approximated near x^0 by the linear function

$f(x) approx log(e^{x_1}+e^{x_2}) + frac{1}{e^{x^0_1}+e^{x^0_2}} left( (x_1-x_1^0) e^{x^0_1} +(x_2-x^0_2) e^{x^0_2}right) .$