What is the derivative of sin(x)?

The derivative of the sine function, sin(x), is cos(x). This is a foundational result in calculus that arises from the limit definition of the derivative as well as from trigonometric limits.

When taking the derivative, we are essentially finding the rate of change of the sine function with respect to the variable x. The cosine function describes how the sine function increases and decreases, aligning perfectly with the geometric interpretation of the unit circle, where the cosine of an angle represents the x-coordinate of a point on the circle while the sine represents the y-coordinate.

Thus, as x increases, sin(x) reaches its maximum value of 1, where its rate of change is 0, corresponding to cos(x) being 0. Conversely, at points where sin(x) is at 0, such as at 0, π, or 2π, the rate of change, or derivative, equals 1 or -1, which corresponds perfectly to cos(x) being either 1 or -1 respectively.

This relationship maintains continuity and smoothness throughout the function, demonstrating that the derivative of sin(x) is indeed cos(x), confirming that the choice is accurate and reflective of the fundamental principles in calculus related to trigonometric functions.