What is the second derivative of #f(x)= ln sqrt(x/e^x)#?
1 Answer
Mar 4, 2017
# f''(x) = -1/(2x^2) #
Explanation:
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
# f'(x) = 1/2(1/x-1) #
Differentiating a second time wrt
# f''(x) = 1/2(-1/x^2) #
# " " = -1/(2x^2) #