NTA Abhyas JEE Main2020MathematicsDifferentiationPractice

If f x = 2 + x 1 + x 1 + x , then f ′ 0 is equal to

Options

A2 log 2
Blog 2
C3 log 2 − 1
D2 log 2 − 1

Correct answer

D. 2 log 2 − 1

Step-by-step solution

By taking logarithm on both sides,   log f ( x )   =   ( 1 + x )   log ( 2 + x ) − log ( 1 + x ) On differentiating, we get, 1 f x · f ' x = 1 + x 2 + x + log ( 2 + x ) − 1 + x 1 + x − log ( 1 + x ) Now, put x = 0 f ' ( 0) = f 0 1 + log 2 2 − 1 = f 0 log 2 − 1 2 = 2 log 2 − 1 ∵ f 0 = 2

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