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Question

Assertion (A):The derivative of (logx)x w.r.t x is (logx)x−1[1+logxlog(logx)]
Reason (R): ddx{f(x)g(x)}=f(x)g(x) (g(x)f′(x)f(x)+g′(x)log(f(x)))

A
Both A and R are true R is correct reason of A
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B
Both A and R are true R is not correct reason of A
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C
A is true but R is false
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D
A is false but R is true
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Solution

The correct option is A Both A and R are true R is correct reason of A
Assertion (logx)x=exlog(logx)
d(logx)xdx=(logx)x(log(log(x))+xlog(x)1x)
(log(x))x1(log(x)log(log(x))+1)
Reason
d(f(x)g(x)))dx=d(eg(x)logf(x))dx
=(f(x))g(x)(g(x)logf(x)+g(x)f(x)f(x))

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