existence of a derivative $x\cdot f(x)$ Lets say we have a function $f(x)$ that has a derivative at point $a$.
Can we prove that the function $x\cdot f(x)$ has also a derivative at point $a$?
If this is not true, can anybody give an example that shows this ...

To avoid using anything but the definition:
Let $g(x)=xf(x)$
$$\lim_{h\to 0} \frac{g(x+h)-g(x)}{h} = \lim_{h\to 0} \frac{(x+h)f(x+h)-xf(x)}{h} = \lim_{h\to 0} \frac{xf(x+h)-xf(x)+hf(x+h)}{h} = x\lim_{h\to 0} \frac{f(x+h)-f(x)}{h} +\lim_{h\to 0}f(x+h) = xf^\prime(x)+f(x)$$
