The linearization of a function at first order is given by:
[tex]f(x+\Delta x)=f(x)+f^{\prime}(x)\times\Delta x\ldots(A)[/tex]
where f'(x) denotes the derivative of f(x). In our case,
[tex]\begin{gathered} f(x)=e^x \\ \text{then} \\ f^{\prime}(x)=\frac{de^x}{dx}=e^x \end{gathered}[/tex]
Then, by substituting this result into equatio (A), we get
[tex]f(x+\Delta x)=e^x+e^x\times\Delta x[/tex]
or equivalently
[tex]f(x+\Delta x)=e^x+\Delta xe^x[/tex]
Therefore, the answer is the first option.
