Differentiate the function y = sin(x).

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

To differentiate y = sin(x), we use the formula for the derivative of a trigonometric function. The derivative of sin(x) is cos(x), so we have:

y' = cos(x)

This means that the slope of the tangent line to the graph of y = sin(x) at any point x is given by cos(x). For example, at x = 0, the slope of the tangent line is cos(0) = 1, so the tangent line is a line with slope 1 that passes through the point (0,0).

We can also use the derivative to find the critical points of the function. A critical point is a point where the derivative is zero or undefined. Since y' = cos(x), the critical points occur when cos(x) = 0. This happens when x = π/2 + nπ or x = -π/2 + nπ, where n is an integer. At these points, the function has a local maximum or minimum.

Finally, we can use the derivative to find the second derivative of the function. The second derivative of y = sin(x) is given by:

y'' = -sin(x)

This means that the concavity of the graph of y = sin(x) changes at the critical points where cos(x) = 0. At these points, the function changes from concave up to concave down or vice versa.