The derivative of the sine function (sin(x)) is the cosine function (cos(x)). In mathematical notation, it can be represented as:

d/dx(sin(x)) = cos(x)

This means that if you have a function y = sin(x), the rate at which y changes with respect to x (its derivative) is given by cos(x).

In simpler terms, when you differentiate the sine function, you get the cosine function. It tells you how the height (y-value) of the sin curve changes as you move along the x-axis. The derivative, cos(x), represents the slope or rate of change of the sin function at any given point.

Here are three examples that demonstrate the application of the derivative of the sine function:

## Example 1: Consider the function y = sin(x).

To find the derivative, we differentiate the sine function with respect to x: dy/dx = d/dx(sin(x)) = cos(x)

So, the derivative of y = sin(x) is dy/dx = cos(x).

## Example 2: Let’s find the derivative of y = 2sin(x).

Again, we differentiate the sine function and multiply by the constant factor: dy/dx = 2 * d/dx(sin(x)) = 2cos(x)

Therefore, the derivative of y = 2sin(x) is dy/dx = 2cos(x).

## Example 3: Consider the function y = sin(2x).

To find the derivative, we differentiate the sine function and apply the chain rule: dy/dx = d/dx(sin(2x)) = cos(2x) * 2

Thus, the derivative of y = sin(2x) is dy/dx = 2cos(2x).

In each example, we differentiate the sine function with respect to x, and the resulting derivative is the cosine function. We can apply additional factors or the chain rule when necessary. The derivative gives us information about the rate of change or slope of the sine function at different points.