What is the derivative of cos(x)?

The derivative of cos(x) is -sin(x). This result arises from the fundamental principles of calculus, specifically from the rules governing the differentiation of trigonometric functions.

To understand why -sin(x) is the derivative, recall that the derivative represents the rate of change of a function with respect to its variable. When considering the cosine function, its graph exhibits a wave-like behavior: it decreases from 1 to -1 as x moves from 0 to π, and then it begins to increase back to 1 as x continues to 2π.

The negative sign in -sin(x) reflects this behavior: as the cosine function reaches its maximum at the peak (0 radians), its instantaneous rate of change (slope) begins at 0, then becomes negative as it descends toward the minimum (π radians), and returns to 0 again as it ascends back.

Thus, the derivative captures the downward slope of the cosine curve, which corresponds to the negative sine function. This relationship showcases how the cosine and sine functions are interconnected through differentiation, reinforcing their positions as foundational elements in trigonometry and calculus.