Differentiate with respect to x:

sin 5x cos 3x

The answer given in the book is : 4 cos 8x + cos 2x

sin 5x cos 3x= 1/2(2 sin 5x cos3x)=1/2(sin 5x+3x + sin 5x-3x) as 2sin a cos b=sin(a+b) + sin(a-b)

=1/2(sin8x + sin 2x)

differentiating,

1/2 d/dx (sin 8x+sin2x)

=1/2 (8cos 8x + 2cos2x)

= 4cos8x + cos 2x