Derivative of cos x. What is the speciﬁc formula for the derivative of the function cos x? This calculation is very similar to that of the derivative of sin(x). If you get stuck on a step here it may help to go back and review the corresponding step there. d As in the calculation of dx sin x, we begin with the deﬁnition of the derivative: d cos(x + Δx) − cos(x) cos x = lim Δx→0 Δx dx Use the angle sum formula cos(a + b) = cos(a) cos(b) − sin(a) sin(b) and then simplify: d cos x dx = = = = d cos x dx = cos x cos Δx − sin x sin Δx − cos x Δx→0 Δx � � cos x cos Δx − cos x − sin x sin Δx lim + Δx→0 Δx Δx � � cos x(cos Δx − 1) − sin x sin Δx lim + Δx→0 Δx Δx � � � � �� cos Δx − 1 sin Δx + (− sin x) lim cos x Δx→0 Δx Δx � � � � cos Δx − 1 sin Δx + lim (− sin x) lim cos x Δx→0 Δx→0 Δx Δx lim Once again we use the following (unproven) facts: cos Δx − 1 Δx sin Δx lim Δx→0 Δx lim Δx→0 = 0 (A) = 1. (B) And we conclude: d cos x dx d cos x dx � = cos Δx − 1 Δx→0 Δx cos x · 0 + (− sin x) · 1 = −sin(x). = lim cos x 1 � � + lim (− sin x) Δx→0 sin Δx Δx �