Question

The left hand derivative of $f\left(x\right)=\left[x\right]sin\pi x$ at $x=k$ is?

(Where k is an integer and [x]=greatest integer $\le x$)

Answer 1

We need to find the left hand derivative of $f\left(x\right)=\left[x\right]sin\pi x$ at $x=k$.

${lim}_{h\to 0^{-}}\frac{f\left(k\right)-f(k-h)}{h}={lim}_{h\to 0^{-}}\frac{\left[k\right]sin\pi k-[k-h]sin\pi (k-h)}{h}$

$={lim}_{h\to 0^{-}}\frac{-(k-1)sin\pi (k-h)}{h}$

We know that, $sink\pi =0\Rightarrow sin(k\pi -\theta )=(-1{)}^{k-1}sin\theta$

$={lim}_{h\to 0^{-}}\frac{-(k-1)(-1{)}^{k-1}sinh\pi }{h\pi }\times \pi$

$=\pi (k-1)(-1{)}^k$

${lim}_{h\to 0^{-}}\frac{f\left(k\right)-f(k-h)}{h}=\pi (k-1)(-1{)}^k$