f(x)=sin(3x)cos(4x),
df/dx=sin(3x)(-4sin(4x))+cos(4x)(3cos(3x))=-4sin(3x)sin(4x)+3cos(3x)cos(4x) or 3cos(3x)cos(4x)-4sin(3x)sin(4x).
sin(A+B)=sinAcosB+cosAsinB, sin(A-B)=sinAcosB-cosAsinB,
sin(A+B)+sin(A-B)=2sinAcosB, sinAcosB=½(sin(A+B)+sinAcosB)
If A=3x and B=4x, sin(A+B)=sin(7x), sin(A-B)=sin(-x)=-sin(x), therefore:
f(x)=½(sin(7x)-sin(x)), df/dx=½(7cos(7x)-cos(x)).
df/dx=3cos(3x)cos(4x)-4sin(3x)sin(4x)=½(7cos(7x)-cos(x)).