What is the second derivative of #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) #