What is the second derivative of $f (x) = ln \sqrt{\frac{x}{e^{x}}}$ $f(x)= ln sqrt(x/e^x)$ ?

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Answer 1 · 2017-03-04T22:03:46

$f''(x) = -1/(2x^2)$

We have:

$f(x)=ln(sqrt(x/e^x))$

which, using the rule of logs, we can write as:

$f(x) = ln((x/e^x)^(1/2))$
$" " = 1/2ln(x/e^x)$
$" " = 1/2{ln(x) - ln(e^x)}$
$" " = 1/2{ln(x) - x}$

Differentiating wrt $x$ gives:

$f'(x) = 1/2(1/x-1)$

Differentiating a second time wrt $x$ gives:

$f''(x) = 1/2(-1/x^2)$
$" " = -1/(2x^2)$

What is the second derivative of f(x)= ln sqrt(x/e^x)f(x)=ln√xexf(x)= ln sqrt(x/e^x)?