Differentiate the function y = cos(x).

The derivative of y = cos(x) is -sin(x).

To differentiate y = cos(x), we use the formula for the derivative of a trigonometric function:

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

This means that the rate of change of the cosine function at any point x is equal to the negative of the sine function at that point.

To see why this is true, we can use the definition of the derivative:

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

Using the trigonometric identity cos(a+b) = cos(a)cos(b) - sin(a)sin(b), we can rewrite the numerator as:

cos(x+h) - cos(x) = cos(x)cos(h) - sin(x)sin(h) - cos(x)

Simplifying this expression and dividing by h, we get:

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

Taking the limit as h approaches 0, we get:

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

Therefore, the derivative of y = cos(x) is -sin(x).