# Differentiation

• March 12th 2013, 03:37 PM
Matt1993
Differentiation
f(x) = (x) / (1 + |x|)
Show from the definition that f: R -> R is differentiable at 0 and find the value of f'(0)

I have been trying to do it from first principles but keep getting no where.

Any help would be appreciated

• March 14th 2013, 07:24 PM
TheEmptySet
Re: Differentiation
Note that

$f(0)=0$

Using the limit definition. Taking the limit from the right gives

$f'(0)=\lim_{h \to 0^+}\frac{\frac{h}{1+h}-0}{h}=\lim_{h \to 0^+}\frac{1}{1+h}=1$

Taking the limit from the left

$f'(0)=\lim_{h \to 0^-}\frac{\frac{h}{1-h}-0}{h}=\lim_{h \to 0^-}\frac{1}{1-h}=1$