Calculate the derivative of sin(x).

The derivative of sin(x) is cos(x).

To find the derivative of sin(x), we use the formula for the derivative of a trigonometric function:

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

This means that the derivative of sin(x) is equal to cos(x). We can prove this using the limit definition of the derivative:

lim h->0 [sin(x+h) - sin(x)]/h

= lim h->0 [sin(x)cos(h) + cos(x)sin(h) - sin(x)]/h

= lim h->0 [cos(h) - 1]/h * sin(x) + lim h->0 cos(x)/h * sin(h)

= 0 * sin(x) + cos(x) * 1

= cos(x)

Therefore, the derivative of sin(x) is cos(x). This means that the slope of the graph of sin(x) at any point x is equal to the value of cos(x) at that point. For example, at x=0, the slope of the graph of sin(x) is equal to cos(0) = 1. At x=π/2, the slope of the graph of sin(x) is equal to cos(π/2) = 0.