Functions/Derivatives

[johnny101]

Suppose that a function f satisfies the following conditions for all real values of x and y.

a: f(x+y)=f(x)*f(y)
b: f(x)=1+xg(x), where the limit x(approaches 0) g(x)=1.

Show f '(x)= f(x).

Can someone show how to do this in steps, specifically the derivative part?

 

[pka]

Suppose that a function f satisfies the following conditions for all real values of x and y.
a: f(x+y)=f(x)*f(y)
b: f(x)=1+xg(x), where the limit x(approaches 0) g(x)=1.
Show f '(x)= f(x).

Can you show that \(\displaystyle f(x + h) = f(x)f(h) = f(x)\left[ {1 + hg(x)} \right]\)?

Use that to find \(\displaystyle f'(x) = {\displaystyle\lim _{h \to 0}}\dfrac{{f(x + h) - f(x)}}{h}\)