Consider the function,
f( x )=sin( x+a )
According to the first principle, the derivative of the function is,
f ′ ( x )= lim h→0 f( x+h )−f( x ) h
Apply the above formula in the given function,
f ′ ( x )= lim h→0 sin( x+h+a )−sin( x+a ) h
From the trigonometric identity,
sinC−sinD=2cos C+D 2 sin C−D 2
The derivative of the given function is,
f ′ ( x )= lim h→0 1 h [ 2cos( x+h+a+x+a 2 )sin( x+h+a−x−a 2 ) ] = lim h→0 1 h [ 2cos( 2x+2a+h 2 )sin( h 2 ) ] = lim h→0 [ cos( 2x+2a+h 2 ) sin h 2 h 2 ] = lim h→0 cos( 2x+2a+h 2 ) lim h 2 →0 sin h 2 h 2
Also we know that,
lim x→0 sinx x =1
Apply the limits,
f ′ ( x )=cos 2x+2a+0 2 ( 1 ) =cos( x+a )
Thus, the derivative of sin( x+a ) is cos( x+a ).