What is the derivative of cos(x)?

The derivative of cos(x) is -sin(x) because it follows from the basic rules of differentiation in calculus. When finding the derivative, you are looking for the rate at which the function changes with respect to x.

The cosine function, which is periodic and oscillates between -1 and 1, has a specific way in which it changes; its maximum values correspond to zero rates of change (i.e., its peak is flat), while it decreases through zero to its minimum value at specific intervals. This behavior is captured in the derivative.

The negative sign arises because the slope of the cosine function decreases as it moves from its maximum (at 0, 2π, etc.) toward zero. Thus, where cos(x) is at a peak, the sine function (which represents the derivative) is zero, and when cos(x) is at zero, sin(x) is at a maximum, indicating a change from increasing to decreasing.

This relationship is crucial in trigonometry and calculus, showing how the derivative functions as the gradient of the original function, providing insights into the original function's behavior.