# Prove g'(x) = g(x)

1. Oct 2, 2012

### drawar

1. The problem statement, all variables and given/known data
Let f and g be two functions on ℝ such that:
1. $g(x) = xf(x) + 1$ for all x,
2. $g(x + y) = g(x)g(y)$ for all x,y,
3. $\mathop {\lim }\limits_{x \to 0} f(x) = 1$.
Prove that: $g'(x) = g(x)$ for all x.

2. Relevant equations

3. The attempt at a solution
Sorry for asking this question without showing my workings but I just don't know how to get started. Any hints would be much appreciated. TIA!

2. Oct 2, 2012

### SammyS

Staff Emeritus
Just use the "limit" definition of the derivative.

3. Oct 2, 2012

### drawar

So I have to prove $\mathop {\lim }\limits_{h \to 0} \frac{{g(x + h) - g(x)}}{h} = g(x)$?

4. Oct 3, 2012

### SammyS

Staff Emeritus
Yes.

It works out quite nicely.

5. Oct 3, 2012

### happysauce

Yup, remember the 3 properties while you use the precise definition of derivative.

6. Oct 3, 2012

### drawar

Thank you all, please check my working below:

$g'(x) = \mathop {\lim }\limits_{h \to 0} \frac{{g(x + h) - g(x)}}{h} = \mathop {\lim }\limits_{h \to 0} \frac{{g(x).g(h) - g(x)}}{h} = \mathop {\lim }\limits_{h \to 0} \frac{{g(x).((g(h) - 1)}}{h} = \mathop {\lim }\limits_{h \to 0} \frac{{g(x)hf(h)}}{h} = \mathop {\lim }\limits_{h \to 0} g(x)f(h) = \mathop {\lim }\limits_{h \to 0} g(x).\mathop {\lim }\limits_{h \to 0} f(h) = \mathop {\lim }\limits_{h \to 0} g(x) = g(x)$

7. Oct 3, 2012

### Crosstalk

That looks correct to me.