Differentiate using logarithmic differentiation.
1)square root of((x+1)(x+2)(x+3))
               3               6
2)sin3x sin  x cos6x  cos x
 
3)tanx tan2x tan3x tan4x
                     x           3x
4)squarerootof(2  sin4x e       )
                  -------------------
                          x+1

Asked by Rajeev | 19th Jul, 2015, 06:15: PM

Expert Answer:

1. C o n s i d e r space t h e space g i v e n space f u n c t i o n y equals square root of open parentheses x plus 1 close parentheses open parentheses x plus 2 close parentheses open parentheses x plus 3 close parentheses end root T a k i n g space log a r i t h m s space o n space b o t h space t h e space s i d e s comma space w e space h a v e comma log y equals log open square brackets square root of open parentheses x plus 1 close parentheses open parentheses x plus 2 close parentheses open parentheses x plus 3 close parentheses end root close square brackets rightwards double arrow l o g y equals l o g open square brackets open parentheses x plus 1 close parentheses open parentheses x plus 2 close parentheses open parentheses x plus 3 close parentheses close square brackets to the power of 1 half end exponent rightwards double arrow l o g y equals 1 half l o g open square brackets open parentheses x plus 1 close parentheses open parentheses x plus 2 close parentheses open parentheses x plus 3 close parentheses close square brackets rightwards double arrow l o g y equals 1 half open square brackets log open parentheses x plus 1 close parentheses plus log open parentheses x plus 2 close parentheses plus log open parentheses x plus 3 close parentheses close square brackets D i f f e r e n t i a t i n g space t h e space a b o v e space f u n c t i o n space w i t h space r e s p e c t space t o space x comma space w e space h a v e comma 1 over y fraction numerator d y over denominator d x end fraction equals 1 half open square brackets fraction numerator 1 over denominator x plus 1 end fraction plus fraction numerator 1 over denominator x plus 2 end fraction plus fraction numerator 1 over denominator x plus 3 end fraction close square brackets rightwards double arrow 1 over y fraction numerator d y over denominator d x end fraction equals 1 half open square brackets fraction numerator x squared plus 5 x plus 6 plus x squared plus 4 x plus 3 plus x squared plus 3 x plus 2 over denominator open parentheses x plus 1 close parentheses open parentheses x plus 2 close parentheses open parentheses x plus 3 close parentheses end fraction close square brackets rightwards double arrow 1 over y fraction numerator d y over denominator d x end fraction equals 1 half open square brackets fraction numerator 3 x squared plus 12 x plus 11 over denominator open parentheses x plus 1 close parentheses open parentheses x plus 2 close parentheses open parentheses x plus 3 close parentheses end fraction close square brackets rightwards double arrow fraction numerator d y over denominator d x end fraction equals y over 2 open square brackets fraction numerator 3 x squared plus 12 x plus 11 over denominator open parentheses x plus 1 close parentheses open parentheses x plus 2 close parentheses open parentheses x plus 3 close parentheses end fraction close square brackets rightwards double arrow fraction numerator d y over denominator d x end fraction equals fraction numerator square root of open parentheses x plus 1 close parentheses open parentheses x plus 2 close parentheses open parentheses x plus 3 close parentheses end root over denominator 2 end fraction open square brackets fraction numerator 3 x squared plus 12 x plus 11 over denominator open parentheses x plus 1 close parentheses open parentheses x plus 2 close parentheses open parentheses x plus 3 close parentheses end fraction close square brackets rightwards double arrow fraction numerator d y over denominator d x end fraction equals 1 half open square brackets fraction numerator 3 x squared plus 12 x plus 11 over denominator square root of open parentheses x plus 1 close parentheses open parentheses x plus 2 close parentheses open parentheses x plus 3 close parentheses end root end fraction close square brackets

Answered by Vimala Ramamurthy | 20th Jul, 2015, 09:09: AM