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In mathematics, the concept of a derivative is central to understanding how functions change at a specific point. However, there are situations where the derivative of a function does not exist at a certain point. This can occur when the function is not smooth or has abrupt changes in behavior at that point.

Consider the example provided: the function is given by \( f(x) = x^{1/3} \), and we are examining its derivative at \( x=0 \). To determine whether the derivative exists, we use the limit definition: \( \ \lim_{{h o 0}} \frac{{f(a+h) - f(a)}}{h} \).

When substituting into the definition, we end up with \( \ \lim_{{h \to 0}} \frac{{h^{1/3}}}{h} = \ \lim_{{h \to 0}} h^{-2/3} \). As \( h \) approaches zero, \( h^{-2/3} \) tends to infinity, which tells us that the limit does not exist. Consequently, the derivative at \( x=0 \) does not exist for this function.

The main takeaway is that a function may not have a derivative at certain points if the behavior of the function becomes non-linear or behaves discontinuously, as seen here.

For a function to be differentiable at a point, it means having a well-defined slope or rate of change at that specific point. Differentiability is closely tied to the existence of a derivative.

In our example of \( f(x) = x^{1/3} \), we're interested in knowing if it's differentiable at \( x=0 \). Using the limit definition confirms that the limit does not exist because it goes to infinity. This implies the function is not differentiable at this point.

Differentiability has a few key requirements:

• The function must be continuous at the point in question.
• There must be no sharp corners or cusps at the point.
• The left-hand derivative and right-hand derivative should be equal.

The cubic root function, \( x^{1/3} \), is continuous at \( x=0 \), but the slope approaches infinity, violating the smoothness needed for differentiability. Thus, smoothness or linearity of change is essential for a function to have a derivative at a given point.

Evaluating limits is a fundamental aspect of calculus and is particularly crucial when using the limit definition of the derivative. In mathematical terms, evaluating a limit explores what happens to a function as the variable approaches a certain value.

For the derivative to exist, the limit \( \ \lim_{{h \to 0}} \frac{{f(a+h) - f(a)}}{h} \) must yield a finite number. If the expression approaches a specific number, then this number is the derivative.

In the case of \( f(x) = x^{1/3} \), the limit \( \ \lim_{{h \to 0}} h^{-2/3} \) does not reach a finite value; instead, it goes to infinity. This means the expression becomes unbounded.

Successfully evaluating the limit would have shown that the function's behavior at a point in terms of how abruptly it changes is manageable and linear. However, when the limit results in infinity or becomes undefined, it indicates non-differentiability at that point.