The right-sided and left-sided derivatives of a function at a point $a$ are given by $f_{+}^{\prime }\left(a\right)=\underset{h\to 0^{+}}{\lim }\frac{f(a+h)-f\left(a\right)}{h}$ and $f_{-}^{\prime }\left(a\right)=\underset{h\to 0^{-}}{\lim }\frac{f(a+h)-f\left(a\right)}{h}$, respectively, provided these limits exist. The derivative $f^{\prime }\left(a\right)$ exists if and only if $f_{+}^{\prime }\left(a\right)=f_{-}^{\prime }\left(a\right)$.
Compute $f_{+}^{\prime }\left(a\right)$ and $f_{-}^{\prime }\left(a\right)$ at the given point $a$.
$f\left(x\right)=\mid x-2\mid$; $a=2$

Verified step by step guidance

1. Identify the function given: \( f(x) = |x - 2| \) and the point \( a = 2 \).
2. To find the right-sided derivative \( f_{+}^{\prime}(2) \), consider \( h \to 0^{+} \). For \( x > 2 \), \( f(x) = x - 2 \). Thus, \( f(2 + h) = (2 + h) - 2 = h \).
3. Compute the right-sided derivative: \( f_{+}^{\prime}(2) = \lim_{h \to 0^{+}} \frac{f(2 + h) - f(2)}{h} = \lim_{h \to 0^{+}} \frac{h - 0}{h} = \lim_{h \to 0^{+}} 1 \).
4. To find the left-sided derivative \( f_{-}^{\prime}(2) \), consider \( h \to 0^{-} \). For \( x < 2 \), \( f(x) = 2 - x \). Thus, \( f(2 + h) = 2 - (2 + h) = -h \).
5. Compute the left-sided derivative: \( f_{-}^{\prime}(2) = \lim_{h \to 0^{-}} \frac{f(2 + h) - f(2)}{h} = \lim_{h \to 0^{-}} \frac{-h - 0}{h} = \lim_{h \to 0^{-}} -1 \).

Key Concepts

Right-sided and Left-sided Derivatives

Right-sided and left-sided derivatives are limits that describe the behavior of a function as it approaches a specific point from the right or left, respectively. The right-sided derivative at a point 'a' is defined as the limit of the difference quotient as 'h' approaches 0 from the positive side, while the left-sided derivative is defined similarly but approaches from the negative side. These derivatives help determine the function's behavior at points where it may not be differentiable.

Existence of the Derivative

The derivative of a function at a point exists if and only if the right-sided and left-sided derivatives at that point are equal. This condition ensures that the function has a well-defined tangent at that point, indicating smoothness and continuity. If the two derivatives are not equal, the function may have a corner or cusp, making it non-differentiable at that point.

Absolute Value Function

The absolute value function, denoted as f(x) = |x - 2|, represents the distance of 'x' from 2 on the real number line. This function is piecewise defined, resulting in a V-shaped graph with a vertex at x = 2. Understanding the behavior of this function is crucial for calculating its derivatives, especially at the point where it changes direction, which is where the left-sided and right-sided derivatives need to be evaluated.