Question

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

• A. $\left[x\right]$
• B. $x-1$
• C. $\left\{x\right\}$
• D. $2\left\{x\right\}$

Answer 1

The correct option is D $2\left\{x\right\}$

$y=x(x-[x\left]\right)-x\left[x\right]y=x^2-x\left[x\right]-x\left[x\right]y=x^2-2x\left[x\right]$
As $x\ne Z,$ Let $\left[x\right]=n,$ where $n\in Z.$
$\frac{dy}{dx}=2x-2n=2(x-[x\left]\right)\therefore \frac{dy}{dx}=2\left\{x\right\}$