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

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Answer 1 · 2016-01-15T14:58:27

$0$

Let $t = sqrt(e^x) = e^(x/2)$

$(dt)/(dx) =1/2*e^(x/2)$

$f(t) = ln t$

$(df)/(dt) = 1/t$

$(df)/(dx) = (df)/(dt)*(dt)/(dx)$

$= 1/t *(e^(x/2))/2 =(e^(x/2))/(2e^(x/2)) = 1/2$

$f''(x) = d/dx(1/2) = 0$

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