Derivative of y=cosx squared?

The derivative of \( y = \cos^2 x \) is \( -2 \cos x \sin x \), which can be equivalently expressed as \( -\sin(2x) \) using the double-angle identity. This result is obtained by applying the chain rule, as the function is a composition of the squaring function and the cosine function. Specifically, if we let \( u = \cos x \), then \( y = u^2 \). The chain rule states that \( dy/dx = (dy/du) \cdot (du/dx) \). Here, \( dy/du = 2u \) and \( du/dx = -\sin x \). Multiplying these gives \( 2 \cos x \cdot (-\sin x) = -2 \cos x \sin x \). The alternative form \( -\sin(2x) \) follows directly from the trigonometric identity \( 2 \sin \theta \cos \theta = \sin(2\theta) \), providing a more compact expression that is often useful in integration or further differentiation.

The choice between the forms \( -2 \cos x \sin x \) and \( -\sin(2x) \) depends on the context of the problem. For instance, if one needs to find critical points by setting the derivative to zero, \( -\sin(2x) = 0 \) simplifies to \( 2x = n\pi \), immediately yielding \( x = n\pi/2 \) for integer \( n \). In contrast, the unsimplified product form might require considering cases where \( \cos x = 0 \) or \( \sin x = 0 \) separately, which is algebraically equivalent but slightly less efficient. This derivative is fundamental in calculus involving trigonometric functions, frequently appearing in problems related to optimization, motion described by harmonic functions, and when integrating expressions like \( \sin(2x) \).

Understanding this differentiation also reinforces the importance of recognizing composite functions and applying identities to simplify results. A common error is to mistakenly apply the power rule alone, yielding \( 2 \cos x \) while neglecting the derivative of the inner function \( \cos x \). The negative sign in the derivative is semantically significant: it indicates that \( \cos^2 x \), which is always non-negative, decreases when \( \cos x \) and \( \sin x \) have the same sign. For example, in the first quadrant where both sine and cosine are positive, the derivative is negative, confirming that \( \cos^2 x \) is decreasing there from its maximum at \( x = 0 \).

In practical applications, this derivative describes the instantaneous rate of change of the squared cosine function, which models physical quantities like intensity in wave optics or power in AC circuits where quantities are proportional to the square of a cosine. The simplified form \( -\sin(2x) \) makes it evident that the rate of change oscillates with twice the frequency of the original function and an amplitude of one, shifted by a negative sign. This characteristic frequency doubling is a direct consequence of the squaring operation, which generates a second harmonic, a pattern that persists in more advanced contexts such as Fourier analysis and signal processing.