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In calculus, the derivative is a core concept that measures how a function changes as its input changes. It's essentially the rate of change of the function with respect to its variable. For a given function \( f(x) \), the derivative at a specific point \( a \) is defined by the limit:\[f'(a) = \lim_{{h \to 0}} \frac{{f(a + h) - f(a)}}{h}\]If this limit exists, it gives us the slope of the tangent line to the function at \( x = a \).

• Derivatives are widely used to find the rate of change in functions such as velocity and acceleration in physics.
• They also help in finding maxima and minima points which are crucial in optimization problems.

For the absolute value function \( f(x) = |3x| \), the problem becomes evaluating this limit as \( h \to 0 \) to check if the function is differentiable at \( x = 0 \). Understanding the derivative through such definitions provides a fundamental basis for more complex mathematical analysis.

Differentiability is a property of a function that indicates whether a derivative exists at each point within its domain. A function is differentiable at a point if it has a defined derivative at that point. To check the differentiability of \( f(x) = |3x| \) at \( x = 0 \), we examine the one-sided limits of the derivative:

• Right-hand limit as \( h \to 0^+ \): \( \lim_{{h \to 0^+}} \frac{{3h}}{h} = 3 \)
• Left-hand limit as \( h \to 0^- \): \( \lim_{{h \to 0^-}} \frac{{-3h}}{h} = -3 \)

If these one-sided limits are equal, the derivative exists, and the function is differentiable at that point. However, since the one-sided limits for \( f(x) = |3x| \) at \( x = 0 \) don't match, the function is not differentiable there. This lack of equality in the limits reflects a corner or cusp in the graph at \( x = 0 \), showing the discontinuity of the derivative.

The absolute value function, represented as \( f(x) = |x| \), outputs the non-negative value of \( x \). It has the peculiar property of changing behavior at \( x = 0 \), which affects its differentiability. For \( f(x) = |3x| \), it's useful to understand how the absolute value impacts derivation. As demonstrated through examination of limits:

• For \( x > 0 \), \( |3x| = 3x \), behaving like a linear function.
• For \( x < 0 \), \( |3x| = -3x \), flipping its direction.

This change in direction at \( x = 0 \) leads to different slopes on either side of zero, causing a sharp corner or cusp. Graphically, this manifests as a V-shape at the origin, indicating the abrupt change in direction and why the derivative doesn't exist at \( x = 0 \). Understanding the behavior of the absolute value function at this critical point reveals how it disrupts the continuity needed for a derivative to exist.