# 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) #