Find out the answer to "Find the derivatives of the functions. s(x)=ln⁡ln⁡x"

naivlingr

Answered question

2021-07-05

Find the derivatives of the functions.

s (x) = \ln \sqrt{\ln x}

avortarF

Skilled2021-07-06Added 113 answers

The function we want to differentiate is

$s (x) = \ln \sqrt{\ln x}$

Let's first simplify the function. Recall that

$\sqrt{\ln x} = (\ln x)^{\frac{1}{2}}$

Also recall that

$\ln (a^{b}) = b \ln a$

Combining the two properties above, we get

$s (x) = \ln (\ln x)^{\frac{1}{2}} = \frac{1}{2} \ln \ln x$

Recall the generalized rule for the derivative of a natural logarithm:

$\frac{d}{d x} [\ln u] = \frac{1}{u} \frac{d u}{d x}$

Now we get

$s^{'} (x) = \frac{1}{2} \frac{1}{\ln x} \frac{d}{d x} (\ln x)$

Recall that

$\frac{d}{d x} (\ln x) = \frac{1}{x}$

Thus

$s^{'} (x) = \frac{1}{2} \frac{1}{\ln x} \frac{1}{x} = \frac{1}{2 x \ln x}$

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