Possible derivation: d/dx(y) = d/dx(1/2 cos(2 x)-x) The derivative of y is zero: 0 = d/dx(-x+1/2 cos(2 x)) Differentiate the sum term by term and factor out constants: 0 = (d/dx(cos(2 x)))/2-d/dx(x) The derivative of x is 1: 0 = 1/2 (d/dx(cos(2 x)))-1 Using the chain rule, d/dx(cos(2 x)) = ( dcos(u))/( du) ( du)/( dx), where u = 2 x and ( d)/( du)(cos(u)) = -sin(u): 0 = -1+1/2-d/dx(2 x) sin(2 x) Factor out constants: 0 = -1-1/2 sin(2 x) 2 d/dx(x) Simplify the expression: 0 = -1-(d/dx(x)) sin(2 x) The derivative of x is 1: Answer: | | 0 = -1-1 sin(2 x)
