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