Given that (d/dx)(sin x) = cos x, find (d/dx)(cos x) without using the definition directly (there are two ways).

both ways use the chain rule, which you had just learned.

first way. since cos(x) = sin(x + pi/2), cos'(x) = sin'(x + pi/2) = cos(x + pi/2) = -sin(x).

second way. since cos^2(x) + sin^2(x) = 1, differentiate both sides to get 2cos(x)cos'(x) + 2sin(x)sin'(x) = 0. sin'(x) we know is cos(x) so to solve the equation, cos'(x) must be -sin(x).