the derivative of $f(x)=|x|^{\frac 52}$ From my point of view answer must be $\frac 52|x|^{\frac 32}$...but answer in my text book is $\frac 52 x |x|^{\frac 12}$....help me to solve it.I have tried it to break the function for positive and negative parts and directly differentiated it but the answer is not matched

Breaking it up into a piecewise function is fine.
$f(x) = x^{\frac 52}$ (for $x \geq 0$) [$1$]
$f(x) = (-x)^{\frac 52}$ (for $x < 0$) [$2$]
Differentiating them separately,
$f'(x) = \frac 52x^{\frac 32}$(for $x \geq 0$) [$1$]
$f'(x) = -\frac 52(-x)^{\frac 32}$(for $x < 0$) [$2$]
The second part can be rearranged to:
$f'(x) = -\frac 52(-x)(-x)^{\frac 12} = \frac 52 x (-x)^{\frac 12} = \frac 52 x|x|^{\frac 12}$, which holds for $x < 0$. Note where the minus signs cancel out.
And for the first piece, a similar rearrangement holds trivially, 
$f'(x) = \frac 52x^{\frac 32} = \frac 52 x (x)^{\frac 12} = \frac 52 x|x|^{\frac 12}$, which holds for $x \geq 0$.
So $f'(x) = \frac 52 x|x|^{\frac 12}$ for the entire domain.
Your book is right.
