second derivative of ln(x)/x

Steps to Solve

1. Find the First Derivative

To find the first derivative of the function ( f(x) = \frac{\ln(x)}{x} ), we will use the quotient rule. The quotient rule states that if you have a function ( \frac{u}{v} ), then the derivative is given by:

$$ f'(x) = \frac{u'v - uv'}{v^2} $$

Here, ( u = \ln(x) ) and ( v = x ). The derivatives are:

• ( u' = \frac{1}{x} )
• ( v' = 1 )

Now applying the quotient rule:

$$ f'(x) = \frac{\left(\frac{1}{x}\right)x - \ln(x)(1)}{x^2} = \frac{1 - \ln(x)}{x^2} $$

2. Find the Second Derivative

Next, we will differentiate ( f'(x) = \frac{1 - \ln(x)}{x^2} ) again using the quotient rule.

Let ( u = 1 - \ln(x) ) and ( v = x^2 ). The derivatives are:

• ( u' = -\frac{1}{x} )
• ( v' = 2x )

Now using the quotient rule again:

$$ f''(x) = \frac{u'v - uv'}{v^2} = \frac{\left(-\frac{1}{x}\right)x^2 - (1 - \ln(x))(2x)}{(x^2)^2} $$

3. Simplify the Second Derivative

Now we simplify the expression:

$$ f''(x) = \frac{-x - 2x(1 - \ln(x))}{x^4} = \frac{-x - 2x + 2x \ln(x)}{x^4} $$

This simplifies to:

$$ f''(x) = \frac{(2\ln(x) - 3)x}{x^4} $$

Rearranging gives:

$$ f''(x) = \frac{2\ln(x) - 3}{x^3} $$