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Question

14. sin (r + a)

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Solution

Consider the function,

f( x )=sin( x+a )

According to the first principle, the derivative of the function is,

f ( x )= lim h0 f( x+h )f( x ) h

Apply the above formula in the given function,

f ( x )= lim h0 sin( x+h+a )sin( x+a ) h

From the trigonometric identity,

sinCsinD=2cos C+D 2 sin CD 2

The derivative of the given function is,

f ( x )= lim h0 1 h [ 2cos( x+h+a+x+a 2 )sin( x+h+axa 2 ) ] = lim h0 1 h [ 2cos( 2x+2a+h 2 )sin( h 2 ) ] = lim h0 [ cos( 2x+2a+h 2 ) sin h 2 h 2 ] = lim h0 cos( 2x+2a+h 2 ) lim h 2 0 sin h 2 h 2

Also we know that,

lim x0 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 ).


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