91Ó°ÊÓ

When we talk about absolute value functions, we refer to expressions encapsulated within absolute value bars, such as the function in our exercise, \( f_{1}(x)=|\sin x| \). The absolute value of a number is its distance from zero on the real number line, without regard to direction. This definition means it is always non-negative.

Absolute value functions have unique characteristics when it comes to differentiability due to their 'V' shaped graph. Differentiability at a point implies that the function is locally linear at that point, which is typically not the case at the vertex of the V. As a result, for the absolute value function, we typically notice that it isn't differentiable at the point where the input is zero because the graph has a sharp turn, or cusp, at this point.

In the context of the exercise, we delve into the specifics of \( f_{1}(x) \) and assess its differentiability, which is further explained in the piecewise functions section.

Let's focus on the derivative of the sine function to understand the solution better. The sine function, \( \sin x \), is one of the fundamental trigonometric functions. It oscillates between -1 and 1 for all real numbers x, and its graph is a smooth, continuous wave.

When we take the derivative of the sine function, we get \( \cos x \), which represents the rate of change of \( \sin x \). The fact that the sine function is differentiable for all x comes from its smoothness; its graph has no sharp turns or cusps. This characteristic continues to be valid when the sine function is modified by taking its absolute value, as seen in \( f_{1}(x)=|\sin x| \), where it maintains differentiability except potentially at points where the sine function switches from positive to negative values or vice versa. This analysis is key to determining the differentiability of the functions addressed in the original exercise.

Piecewise functions like \( f_{1}(x)=|\sin x| \) and \( f_{2}(x)=\sin |x| \) are defined by different expressions on different intervals of the domain. They are particularly interesting when it comes to differentiability because each 'piece' of the function may behave differently.

For example, \( f_{1}(x) \) is derived by considering separate expressions for \( \sin x \) when it is positive and negative. On the other hand, \( f_{2}(x) \) is differentiated by separating cases where x itself is positive and negative, leading to its piecewise form. The derivatives of both functions coincide with the derivative of \( \sin x \) where the functions are smooth, but we hit a snag at the boundaries of the pieces.

In particular, the differentiability of \( f_{2}(x) \) at \( x=0 \) is problematic because the behavior of \( \sin |x| \) changes abruptly at this point. Exactly at \( x=0 \) we can't find a common derivative, as on the left we have a negative slope, and on the right, a positive. Hence, \( f_{2}(x) \) is non-differentiable at exactly \( x=0 \) due to this abrupt change in direction – the hallmark of a non-differentiable point in a piecewise function.