I'm using Frechet derivative to get derivative of operator
$$ {\mathbf{F}}{'_{ij}}\left( {{{\mathbf{U}}^{\left( k \right)}}} \right)\delta {U_j} = \mathop {\lim }\limits_{\eta \to 0} \frac{{{F_i}\left( {{{\mathbf{U}}^{\left( k \right)}} + \eta \delta {U_j}} \right) - {F_i}\left( {{{\mathbf{U}}^{\left( k \right)}}} \right)}}{\eta } $$
The function in question is this $$ F_2(u,v)=\frac{\varepsilon }{{qN{\mu _n}}}\left( {{e^{u - v}}\frac{\partial }{{\partial t}}\left( {u - v} \right) + \frac{{{e^{u - v}} }}{{{\tau _n}}}} \right) - \frac{\partial }{{\partial x}}\left\{ {{e^{u - v}}\frac{{\partial v}}{{\partial x}}} \right\} $$
And the frechet derivative I want is $$ F_2'(u)\delta u $$
Here is what i did $$ \mathop {\lim }\limits_{\eta \to 0} \frac{{\frac{\varepsilon }{{qN{\mu _n}}}\left( {{e^{u + \eta \delta {u^{\left( k \right)}} - v}}\frac{\partial }{{\partial t}}\left( {u + \eta \delta {u^{\left( k \right)}} - v} \right) + \frac{{{e^{u + \eta \delta {u^{\left( k \right)}} - v}}}}{{{\tau _n}}}} \right) - \frac{\partial }{{\partial x}}\left\{ {{e^{u + \eta \delta {u^{\left( k \right)}} - v}}\frac{{\partial v}}{{\partial x}}} \right\} - \frac{\varepsilon }{{qN{\mu _n}}}\left( {{e^{u - v}}\frac{\partial }{{\partial t}}\left( {u - v} \right) + \frac{{{e^{u - v}}}}{{{\tau _n}}}} \right) + \frac{\partial }{{\partial x}}\left\{ {{e^{u - v}}\frac{{\partial v}}{{\partial x}}} \right\}}}{\eta } $$ $$ = \mathop {\lim }\limits_{\eta \to 0} \frac{{\frac{\varepsilon }{{qN{\mu _n}}}\left( {{e^{u - v}}\left\{ {{e^{\eta \delta {u^{\left( k \right)}}}}\left[ {\frac{\partial }{{\partial t}}\left( {u - v} \right) + \frac{\partial }{{\partial t}}\left( {\eta \delta {u^{\left( k \right)}}} \right)} \right] - \frac{\partial }{{\partial t}}\left( {u - v} \right)} \right\} + \frac{{{e^{u + \eta \delta {u^{\left( k \right)}} - v}}}}{{{\tau _n}}} - \frac{{{e^{u - v}}}}{{{\tau _n}}}} \right) - \frac{\partial }{{\partial x}}\left\{ {{e^{u + \eta \delta {u^{\left( k \right)}} - v}}\frac{{\partial v}}{{\partial x}}} \right\} + \frac{\partial }{{\partial x}}\left\{ {{e^{u - v}}\frac{{\partial v}}{{\partial x}}} \right\}}}{\eta } $$ $$ = \frac{\varepsilon }{{qN{\mu _n}}}{e^{u - v}}\left\{ {\delta {u^{\left( k \right)}}\frac{\partial }{{\partial t}}\left( {u - v} \right) + \frac{\partial }{{\partial t}}\left( {\delta {u^{\left( k \right)}}} \right) + \frac{{\delta {u^{\left( k \right)}}}}{{{\tau _n}}}} \right\} - \frac{\partial }{{\partial x}}\left\{ {{e^{u - v}}\delta {u^{\left( k \right)}}\frac{{\partial v}}{{\partial x}}} \right\} $$
Is this correct?