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

$f(x)=|x|^\frac 5 2$.
Using power rule and chain rule:
$f'(x)=\frac 5 2  |x|^\frac 3 2 * {d\over dx} |x|)$. Now we have to find ${d\over dx} |x|$. $|x|$ equals $-x$ if $x<0$, and $|x|=x$ if $x\ge0$, from here it is easy to see that the derivative of $|x|$ is uqual to ${x \over |x|}$. We put this in the expression above and we get:
$f'(x)=\frac 5 2  |x|^\frac 3 2 * {x \over |x|}=\frac 5 2  |x|^\frac 1 2 * x$. Note that $x$ can not equal $0$, because the derivative of $|x|$ is undefined at $x=0$.
