Sine and cosine gradients

ddxsin x=cos x, ddxcos x=-sin x for radian arguments. The derivative must match the original curve’s changing slope: sine rises through zero with gradient one; cosine has a horizontal maximum at zero. The gradient curves are quarter-cycle shifts. Black is sine; blue is its gradient, cosine. Where sine rises/falls, the cosine curve lies above/below the input axis. Method Differentiate the outer sine or cosine, include the minus sign for cosine, then multiply by the inner derivative. Example y=cos(1-2x), y =-sin(1-2x)(-2)=2sin(1-2x) Exam check These rules use radians. For a negative inner gradient, track both minus signs instead of losing one.