Left Hand Derivative (LHD) & Right Hand Derivative (RHD) of a Function

Understanding the concepts of Left Hand Derivative (LHD) and Right Hand Derivative (RHD) is crucial for analyzing the behavior of functions, especially at points where they may not be differentiable in the usual sense. These concepts are particularly important when dealing with functions that have sharp turns or corners at certain points.

Definitions

The derivative of a function at a point gives us the slope of the tangent to the function at that point. However, at certain points, the usual definition of a derivative may not apply, such as at corners or cusps. This is where LHD and RHD come into play.

Left Hand Derivative (LHD)

The Left Hand Derivative of a function $f(x)$ at a point $x = a$ is defined as the limit of the average rate of change of the function as the change in $x$ approaches 0 from the left side of $a$. Mathematically, it is expressed as:

$$ LHD = \lim_{h \to 0^-} \frac{f(a + h) - f(a)}{h} $$

Right Hand Derivative (RHD)

Similarly, the Right Hand Derivative of a function $f(x)$ at a point $x = a$ is the limit of the average rate of change of the function as the change in $x$ approaches 0 from the right side of $a$. It is given by:

$$ RHD = \lim_{h \to 0^+} \frac{f(a + h) - f(a)}{h} $$

Differences and Important Points

Aspect	Left Hand Derivative (LHD)	Right Hand Derivative (RHD)
Definition	Limit as $h$ approaches 0 from the left	Limit as $h$ approaches 0 from the right
Mathematical Expression	$\lim_{h \to 0^-} \frac{f(a + h) - f(a)}{h}$	$\lim_{h \to 0^+} \frac{f(a + h) - f(a)}{h}$
Interpretation	Slope of the tangent from the left side	Slope of the tangent from the right side
Differentiability Criteria	If LHD exists and is finite	If RHD exists and is finite
Continuity Requirement	Not necessarily continuous from the left	Not necessarily continuous from the right

Examples

Let's look at some examples to illustrate LHD and RHD.

Example 1: Differentiable Function

Consider the function $f(x) = x^2$. The derivative of this function at any point is $2x$. Let's calculate the LHD and RHD at $x = 1$.

• LHD at $x = 1$:

$$ \begin{align*} LHD &= \lim_{h \to 0^-} \frac{(1 + h)^2 - 1^2}{h} \ &= \lim_{h \to 0^-} \frac{1 + 2h + h^2 - 1}{h} \ &= \lim_{h \to 0^-} \frac{2h + h^2}{h} \ &= \lim_{h \to 0^-} (2 + h) \ &= 2 \end{align*} $$

• RHD at $x = 1$:

$$ \begin{align*} RHD &= \lim_{h \to 0^+} \frac{(1 + h)^2 - 1^2}{h} \ &= \lim_{h \to 0^+} \frac{1 + 2h + h^2 - 1}{h} \ &= \lim_{h \to 0^+} \frac{2h + h^2}{h} \ &= \lim_{h \to 0^+} (2 + h) \ &= 2 \end{align*} $$

Since both LHD and RHD are equal and finite, the function is differentiable at $x = 1$.

Example 2: Non-Differentiable Function

Consider the function $f(x) = |x|$. This function has a sharp turn at $x = 0$. Let's calculate the LHD and RHD at $x = 0$.

• LHD at $x = 0$:

$$ \begin{align*} LHD &= \lim_{h \to 0^-} \frac{|0 + h| - |0|}{h} \ &= \lim_{h \to 0^-} \frac{-h}{h} \ &= \lim_{h \to 0^-} -1 \ &= -1 \end{align*} $$

• RHD at $x = 0$:

$$ \begin{align*} RHD &= \lim_{h \to 0^+} \frac{|0 + h| - |0|}{h} \ &= \lim_{h \to 0^+} \frac{h}{h} \ &= \lim_{h \to 0^+} 1 \ &= 1 \end{align*} $$

Since LHD and RHD are not equal, the function is not differentiable at $x = 0$.

Conclusion

The concepts of Left Hand Derivative and Right Hand Derivative are essential for understanding the behavior of functions at points where the usual derivative may not exist. By examining the LHD and RHD, we can determine whether a function is differentiable at a given point and understand the nature of its graph around that point.