Ex 5.5, 16 - Find derivative of f(x)=(1+x)(1+x2)(1+x4)(1+x8) - Ex 5.5

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Ex 5.5, 16 Find the derivative of the function given by f ( ) = (1 + ) (1 + 2 ) (1 + 4 ) (1 + 8) and hence find f (1) . Given = 1+ 1+ 2 1+ 4 1+ 8 Let = 1+ 1+ 2 1+ 4 1+ 8 Taking log both sides log = log 1+ 1+ 2 1+ 4 1+ 8 log = log 1+ + log 1+ 2 + log 1+ 4 + log 1+ 8 Differentiating both sides . . . . log = log 1 + + log 1 + 2 + log 1 + 4 + log 1 + 8 log = log 1 + + log 1 + 2 + log 1 + 4 + log 1 + 8 log . = 1 1 + . 1 + + 1 1 + 2 . 1 + 2 + 1 1 + 4 . 1 + 4 + 1 1 + 8 . 1 + 8 1 . = 1 1 + . 0+1 + 1 1 + 2 . 0+2 s + 1 1 + 4 . 0+4 3 + 1 1 + 8 . 0+8 7 1 . = 1 1 + + 2 1 + 2 + 4 3 1 + 4 + 8 7 1 + 8 = 1 1 + + 2 1 + 2 + 4 3 1 + 4 + 8 7 1 + 8 = 1+ 1+ 2 1+ 4 1+ 8 1 1 + + 2 1 + 2 + 4 3 1 + 4 + 8 7 1 + 8 Hence, = + + + + + + + + + + + We need to find 1 Putting =1 1 = 1+1 1+ 1 2 1+ 1 4 1+ (1) 8 1 1 +1 + 2 1 1+ 1 2 + 4 1 3 1 + 1 4 + 8 1 7 1 + 1 8 = 2 1+1 1+1 1+1 1 1 + 1 + 2 1 + 1 + 4 1 + 1 + 8 1 + 1 = 2 2 2 2 1 2 + 2 2 + 4 2 + 8 2 = 16 1 + 2 + 4 + 8 2 = 16 15 2 = 8 15 = 120

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