What is the derivative of sin(x)

The derivative of sin(x) is cos(x) because, from the limit definition, [sin(x+h) − sin(x)]/h simplifies to sin(x)(cos h − 1)/h + cos(x)(sin h)/h, which tends to 0 + cos(x) as h → 0. Geometrically, the slope of the sine curve at each x equals the cosine value: at x = 0 the slope is 1 (cos 0 = 1), and at x = π/2 the slope is 0 (cos π/2 = 0). Memory tip: derivatives of trig functions cycle every 90° (π/2): sin → cos → −sin → −cos → sin. Another cue: sine’s derivative keeps the “positive” sign first (sin’ = cos), while cosine’s derivative picks up a minus (cos’ = −sin).