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

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- Mar 12th 2013, 03:37 PMMatt1993Differentiationf(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

- Mar 14th 2013, 07:24 PMTheEmptySetRe: Differentiation
Note that

$\displaystyle f(0)=0$

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

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

Taking the limit from the left

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