Answer
Using Eq. (5) of Section 15.4, we show that the linear approximation in terms of the change in $f$ can be written as
$\Delta f \simeq \nabla {f_p}\cdot\Delta {\bf{v}}$
Work Step by Step
Let $x=a+h$ and $y=b+k$.
Using Eq. (5) of Section 15.4, the linear approximation in terms of the change in $f$ can be written as
$\Delta f \simeq {f_x}\left( {a,b} \right)h + {f_y}\left( {a,b} \right)k$
But $\nabla {f_p} = \left( {{f_x}\left( {a,b} \right),{f_y}\left( {a,b} \right)} \right)$ and $\Delta {\bf{v}} = \left( {h,k} \right)$. Thus,
$\Delta f \simeq \nabla {f_p}\cdot\Delta {\bf{v}}$
