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The calculation of a derivative is a fundamental process in calculus that measures how a function changes as its input changes. To find the derivative of a function at a particular point, say at \(x = 0\), we calculate the limit of the difference quotient. For a function \(f(x)\), this is mathematically expressed as:\[f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}.\]In the given problem, where \(f(x) = x \cdot |x|\), we need to consider two cases because the behavior of \(|x|\) changes around zero. When \(x\) is non-negative, \(f(x)\) simplifies to \(x^2\), and when \(x\) is negative, it becomes \(-x^2\). These variations in the function's definition necessitate analyzing it separately from the left and right of \(x = 0\). By doing this, we determine that both derivatives calculated from either side converge to the same value at \(x = 0\). This matching result confirms the differentiability of the function at the point of interest.

Understanding the behavior of a function around a specific point helps predict its differentiability. The function \(f(x) = x \cdot |x|\) behaves differently based on the sign of \(x\). - For \(x \, \geq \, 0\), \(f(x)\) takes the form \(x^2\), which is a standard parabolic curve opening upwards. - For \(x \, < \, 0\), \(f(x)\) becomes \(-x^2\), representing an upside-down parabola.At \(x = 0\), these two pieces meet smoothly because both give the value zero at this point. This continuity in value at \(x = 0\) hints at potential differentiability, provided the derivatives from either side also match. Observing these behaviors graphically or analytically ensures a better understanding of how changes on one side of the point relate to changes on the other, which is crucial for mastering the topic of differentiability.

The left-hand derivative is the limit of the derivative as we approach a point from the negative side. Mathematically, for a function \(f(x)\) at a point \(x = 0\), the left-hand derivative is expressed as:\[f'_{-}(0) = \lim_{h \to 0^-} \frac{f(0 + h) - f(0)}{h}.\]In our case, when approaching zero from the left (for \(x < 0\)), \(f(x) = -x^2\). Using the definition, the calculation simplifies: \[f'_{-}(0) = \lim_{h \to 0^-} \frac{-h^2 - 0}{h} = \lim_{h \to 0^-} (-h) = 0.\]This result indicates that the rate of change of \(f(x)\) as we approach zero from the left is zero. Understanding left-hand derivatives helps in confirming whether the function's rate of change is consistent across a given point from the negative side.

The right-hand derivative explores how a function changes as its input approaches a given point from the positive direction. For a function \(f(x)\) at \(x = 0\), it is represented by:\[f'_{+}(0) = \lim_{h \to 0^+} \frac{f(0 + h) - f(0)}{h}.\]In this exercise, for \(x \geq 0\), the function is \(f(x) = x^2\). Applying the definition gives:\[f'_{+}(0) = \lim_{h \to 0^+} \frac{h^2 - 0}{h} = \lim_{h \to 0^+} h = 0.\]This indicates the function is not changing as we move to zero from the right side; its rate of change is also zero, just like from the left side. Both the left-hand and right-hand derivatives being equal is crucial in verifying a function's differentiability at a certain point. In this scenario, the consistency on both sides confirms that \(f(x) = x \cdot |x|\) is differentiable at \(x = 0\).