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Question: 27. Derivative of $log_{e^2}(log x)$ w.r.t. to x...

  1. Derivative of loge2(logx)log_{e^2}(log x) w.r.t. to x
A

2xlogx\frac{2}{x log x}

B

1xlogx\frac{1}{x log x}

C

1xlogx2\frac{1}{x log x^2}

D

2logx\frac{2}{log x}

Answer

c) 1xlogx2\frac{1}{x log x^2}

Explanation

Solution

We have

f(x)=log⁡e2(log⁡x)=ln⁡(log⁡x)ln⁡(e2)=ln⁡(log⁡x)2.f(x)=\log_{e^2}(\log x)=\frac{\ln(\log x)}{\ln(e^2)}=\frac{\ln(\log x)}{2}.

Differentiate using the chain rule:

f′(x)=12⋅1log⁡x⋅ddx(log⁡x)=12⋅1log⁡x⋅1x=12x log⁡x.f'(x)=\frac{1}{2}\cdot\frac{1}{\log x}\cdot\frac{d}{dx}(\log x)=\frac{1}{2}\cdot\frac{1}{\log x}\cdot\frac{1}{x}=\frac{1}{2x\,\log x}.

Notice that

12x log⁡x=1x (2log⁡x)=1xlog⁡x2,\frac{1}{2x\,\log x}=\frac{1}{x\,(2\log x)}=\frac{1}{x\log x^2},

since log⁡x2=2log⁡x\log x^2=2\log x.

Thus, the answer is Option (c).