Decide if the function is differentiable at x=0 . Try...

Decide if the function is differentiable at $x=0 .$ Try zooming in on a graphing calculator, or calculating the derivative $f^{\prime}(0)$ from the definition.
$$f(x)=\left\{\begin{array}{ll}
-2 x & \text { for } x<0 \\
x^{2} & \text { for } x \geq 0
\end{array}\right.$$

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Key Concepts

Derivative Definition

The derivative at a point is formally defined as the limit of the difference quotient as the increment approaches zero. This limit, if it exists, provides the instantaneous rate of change or slope of the function at that point, which is a fundamental concept in differential calculus.

One-Sided Derivatives

One-sided derivatives are the limits of the difference quotient as the variable approaches the point of interest from the left (left-hand derivative) and from the right (right-hand derivative). For a function to be differentiable at a point, these one-sided derivatives must exist and be equal.

Piecewise Functions

Piecewise functions are defined by different formulas over different portions of their domains. When analyzing differentiability, it is crucial to focus on the points where the function's definition changes, as these are common locations for potential discontinuities or non-smooth behavior.

Differentiability

Differentiability refers to the quality of a function to have a well-defined and unique tangent (or instantaneous rate of change) at a specific point. This involves checking that the derivative exists at that point, meaning the function behaves in a locally linear manner around that point.

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