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Question

If f(x)=a+xb+xa+b+2x then, f'(0)=


A

2logab+b2-a2ab

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B

aba+b2logab+b2-a2ab

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C

aba+bb2-a2ab

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D

None of these

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Solution

The correct option is B

aba+b2logab+b2-a2ab


Explanation for the correct option.

Step 1: Find the first derivative.

Let, y=f(x), so y=a+xb+xa+b+2x.

Taking log on both sides we get,

logy=a+b+2xloga+xb+x

By differentiating with respect to x we get,

1ydydx=2loga+xb+x+a+b+2x×1a+xb+x×b+x1-a+x1b+x21ydydx=2loga+xb+x+a+b+2x×b+xa+x×b-ab+x21ydydx=2loga+xb+x+a+b+2xb-aa+xb+xdydx=y2loga+xb+x+a+b+2xb-aa+xb+x...(1)

Step 2: Find f'(0).

Now, dydx=f'x and y=f(x). So, 1 can be written as

f'x=a+xb+xa+b+2x2loga+xb+x+a+b+2xb-aa+xb+x

By putting x=0, we get

f'0=aba+b2logab+a+bb-aab=aba+b2logab+b2-a2ab

Hence, option B is correct.


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