The Derivative of the Signum Function Applied to Sine: A Comprehensive Overview

Abstract: This article delves into the complex relationship between the signum function, denoted as sgnx, and the sine function, sinx. Specifically, it addresses the derivative of the composite function sgn(sinx) and explores the underlying principles and properties of these functions. The discussion includes a detailed analysis of the piecewise constant behavior and discontinuous nature of the resulting function, along with step-by-step mathematical proofs for a comprehensive understanding.

What is the Derivative of sgnx Applied to sinkx?

The derivative of the signum function applied to the sine function, denoted as sgn(sinx), is a fascinating topic in calculus. Let's explore this in detail.

The signum function, denoted as sgnx, is defined as:

sgnx { -1 if x 0, 0 if x 0, 1 if x 0 }

When applying the signum function to the sine function, we have:

sgn(sinx) { -1 if sinx 0, 0 if sinx 0, 1 if sinx 0 }

The derivative of this piecewise constant function is undefined at points where sinx 0. Mathematically, the derivative is calculated as:

[ frac{d}{dx}sgn(sinx) 0, ] everywhere except at points where sinx 0 (i.e., x npi where nin mathbb{Z}).

This discontinuity arises because the signum function returns a constant value and hence the function does not change continuously. The jumps in the signum function cancel out the oscillations of the sine function, resulting in a derivative of 0 everywhere except at the points of discontinuity.

Proving the Derivative of sinx

To prove the derivative of the sine function sinx from first principles, we start with the definition of a derivative:

[ frac{d}{dx}f(x) lim_{hto 0}frac{f(x h)-f(x)}{h} ]

Applying this to sinx we get:

[ frac{d}{dx}sin x lim_{hto 0}frac{sin(x h)-sin x}{h} ]

Using the trigonometric identity ( sin(a) - sin(b) 2cosleft(frac{a b}{2}right)sinleft(frac{a-b}{2}right) ), we can rewrite the expression:

[ frac{d}{dx}sin x lim_{hto 0}frac{2cosleft(frac{x h x}{2}right)sinleft(frac{h}{2}right)}{h} ]

Further simplifying:

[ frac{d}{dx}sin x lim_{hto 0}cosleft(frac{2x h}{2}right)lim_{hto 0}frac{sinleft(frac{h}{2}right)}{frac{h}{2}} ]

Since it is known that ( lim_{thetato 0}frac{sintheta}{theta} 1 ), we can simplify the above expression to:

[ frac{d}{dx}sin x cos x ]

Behavior of the Function sgn(sinx)

The function sgn(sinx) has specific behaviors based on the value of sinx.

For sinx0, sgn(sinx)1. For sinx0, sgn(sinx)-1. For sinx0, sgn(sinx)0.

Due to the periodic nature of the sine function, the signum function will also exhibit periodic behavior. It is important to note that the derivative is undefined at points where the sine function crosses zero because at these points, the function is not continuous.

To summarize, the derivative of sgn(sinx) is zero everywhere except at the points where sinx0, i.e., at multiples of pi. At these points, the function is not differentiable.