Question

The answer to the question is one of the below

Answer 1

The linearization of a function at first order is given by:

`f(x+\Delta x)=f(x)+f^(\prime)(x)*\Delta x\ldots(A)`

where f'(x) denotes the derivative of f(x). In our case,

`\begin{gathered} f(x)=e^x \\ \text{then} \\ f^(\prime)(x)=(de^x)/(dx)=e^x \end{gathered}`

Then, by substituting this result into equatio (A), we get

`f(x+\Delta x)=e^x+e^x*\Delta x`

or equivalently

`f(x+\Delta x)=e^x+\Delta xe^x`

Therefore, the answer is the first option.