Question

sin x +coS x17sin x-cosx

Answer 1

Consider the function,

f( x )= sinx+cosx sinx−cosx

The quotient rule of derivative to find the derivative of the function is,

d dx ( U V )= ( U V ′ −V U ′ ) V 2

Where U ′ and V ′ are the derivative of their respective functions.

Apply above formula in the given function,

f ′ ( x )= ( sinx−cosx ) d dx ( sinx+cosx )−( sinx+cosx ) d dx ( sinx−cosx ) ( sinx−cosx ) 2 = ( sinx−cosx )[ d dx sinx+ d dx cosx ]−( sinx+cosx )[ d dx sinx− d dx cosx ] ( sinx−cosx ) 2 = ( sinx−cosx )( cosx−sinx )−( sinx+cosx )( cosx+sinx ) ( sinx−cosx ) 2 = − ( sinx−cosx ) 2 − ( sinx+cosx ) 2 ( sinx−cosx ) 2

Simplify further,

f ′ ( x )= −[ sin 2 x+ cos 2 x−2sinxcosx+ sin 2 x+ cos 2 x+2sinxcosx ] ( sinx−cosx ) 2 = −( 1+1 ) ( sinx−cosx ) 2 = −2 ( sinx−cosx ) 2

Thus, the derivative of sinx+cosx sinx−cosx is −2 ( sinx−cosx ) 2 .