What is the derivative of sin(x)?

The derivative of sin(x) is cos(x) due to the fundamental rules of differentiation in calculus. When applying the derivative to the sine function, we utilize the formal definition of the derivative along with trigonometric identities.

The process can be understood through the limit definition of a derivative. When we take the limit of the difference quotient for sin(x), we find that as we approach the limit, the result simplifies to cos(x). This relationship is rooted in the properties of the unit circle and how the sine and cosine functions are defined as the vertical and horizontal coordinates of points on the circle, respectively.

Additionally, the behavior of these functions around key points shows that the rate of change of sin(x) correlates to the value of cos(x) at any given point along the sine curve. Therefore, the correct answer clearly reflects this established relationship within differential calculus and trigonometric functions.