Answer:
See Below.
Step-by-step explanation:
We want to show that the function:
[tex]f(x) = e^x - e^{-x}[/tex]
Increases for all values of x.
A function is increasing whenever its derivative is positive.
So, find the derivative of our function:
[tex]\displaystyle f'(x) = \frac{d}{dx}\left[e^x - e^{-x}\right][/tex]
Differentiate:
[tex]\displaystyle f'(x) = e^x - (-e^{-x})[/tex]
Simplify:
[tex]f'(x) = e^x+e^{-x}[/tex]
Since eˣ is always greater than zero and e⁻ˣ is also always greater than zero, f'(x) is always positive. Hence, the original function increases for all values of x.
