Answer:
[tex]f(x)=e^x[/tex]
Step-by-step explanation:
Proof:
Let [tex]e^x=z[/tex]
take logs in both sides[tex]ln(e^x)=ln(z)[/tex]
using log properties we have [tex]xln(e)=ln(z)[/tex]. We know that [tex]ln(e)=1[/tex] then [tex]x=ln(z)[/tex]
Taking derivatives with respect to x, we have: [tex]1=\frac{1}{z}\frac{dz}{dx}[/tex]
Next we can move things around [tex]z=\frac{dz}{dx}[/tex]
Replacing z for [tex]e^x[/tex] as defined in the first line, we get
[tex]e^x=\frac{d}{dx}e^x[/tex]
