Find the derivative of sin(x)

For an angle measured in radians, the derivative is cos(x). The limit definition shows where this result comes from.

Step 1: Start with the limit definition

For a function f(x), the derivative is f′(x) = lim(h→0) [f(x+h) − f(x)] / h. With f(x) = sin(x), this becomes lim(h→0) [sin(x+h) − sin(x)] / h.

Step 2: Expand sin(x+h)

The angle addition identity gives sin(x+h) = sin(x)cos(h) + cos(x)sin(h). Substitute it and rearrange:

[sin(x+h) − sin(x)] / h = sin(x)[cos(h) − 1]/h + cos(x)sin(h)/h.

Step 3: Evaluate the two limits

In radians, lim(h→0) [cos(h) − 1]/h = 0 and lim(h→0) sin(h)/h = 1. Therefore,

d/dx sin(x) = sin(x)·0 + cos(x)·1 = cos(x).

For example, at x = π/3, the slope of the sine curve is cos(π/3) = 1/2.