Animated Solution for Mathematics - Limits, Continuity and Differentiability: The left-hand derivative of $f(x)=[x]\sin (\pi x)$ at $x=k,k$ an integer, is

We are investigating the function $f(x)=[x]\sin (\pi x)$ and seeking its left-hand derivative at an integer $k$. While the product rule is a standard tool, the greatest integer function $[x]$ is non-differentiable at integers, necessitating a limit-based approach.

The left-hand derivative is defined as:

First, we evaluate $f(k)$. Since $k$ is an integer, $\sin (k\pi )=0$, which implies $f(k)=[k]\cdot 0=0$. This simplifies our limit to:

Consider the term $f(k-h)$ where $h$ is a small positive value. This places $k-h$ in the interval $(k-1,k)$.

In this specific interval, the greatest integer function is constant: $[k-h]=k-1$. Consequently, the function simplifies to:

We now expand the sine term using the identity $\sin (A-B)=\sin A\cos B-\cos A\sin B$:

Since $\sin (k\pi )=0$ and $\cos (k\pi )=(-1{)}^k$, the expression becomes:

Substituting this back into our limit definition, we obtain:

We can factor out the constants and utilize the standard limit ${\lim }_{\theta \to 0}\frac{\sin \theta }{\theta }=1$:

Multiplying the numerator and denominator by $\pi$, we find:

The final result for the left-hand derivative at integer $k$ is:

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