Find the left- and right-hand derivatives of each of the...

Find the left- and right-hand derivatives of each of the following at the point indicated:
(a) $f(x)=x-|x| \quad$ at $x-2$
(b) $f(x)=\left\{\begin{array}{ll}\frac{x}{1+\mathrm{e}^{1 / x}} & \text { if } x \neq 0 \\ 0 & \text { if } x=0\end{array}\right.$ at $x=0$

Key Concepts

Differentiability of Piecewise Functions

Piecewise functions are defined by different expressions across various intervals. At points where the definition changes, it is crucial to verify both the continuity of the function and the equality of the left-hand and right-hand derivatives. Discrepancies in the derivatives can indicate a corner or cusp, meaning the function is not differentiable at that point.

Absolute Value Function Differentiation

Functions that involve the absolute value often have a change in behavior depending on the sign of the input. When differentiating such functions, it is common to split the analysis into cases where the internal expression is positive or negative. This case analysis is essential to properly calculate the left-hand and right-hand derivatives, especially at the point where the expression inside the absolute value changes sign.

Definition of Derivative

The derivative of a function at a point is defined as the limit of the difference quotient as the increment approaches zero. This limit represents the instantaneous rate of change or the slope of the tangent line at that point.

One-Sided Derivatives

One-sided derivatives are the limits of the difference quotient taken as the variable approaches the point of interest from one side only. The left-hand derivative considers limits from values smaller than the point, while the right-hand derivative considers values larger than the point. For a function to be differentiable at that point, these one-sided derivatives must exist and be equal.

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