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Even Function

If y=xx-x[x],...

Question

If $y = x {x} - x [x],$ then $\frac{d y}{d x}$ is equal to
$($ Where $x \neq Z, [\cdot]$ denotes the greatest integer function and ${\cdot}$ denotes the fractional part function $)$

A

[x]

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B

x - 1

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C

{x}

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D

2 {x}

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Solution

The correct option is D $2 {x}$
$y = x (x - [x]) - x [x] y = x^{2} - x [x] - x [x] y = x^{2} - 2 x [x]$
As $x \neq Z,$ Let $[x] = n,$ where $n \in Z .$
$\frac{d y}{d x} = 2 x - 2 n = 2 (x - [x]) ∴ \frac{d y}{d x} = 2 {x}$

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