second derivative of ln(x)/x
Understand the Problem
The question is asking for the second derivative of the function ln(x)/x. To solve this, we will first differentiate the function to find the first derivative and then differentiate the first derivative to find the second derivative.
Answer
The second derivative is given by \( f''(x) = \frac{2\ln(x) - 3}{x^3} \).
Answer for screen readers
The second derivative of the function ( \frac{\ln(x)}{x} ) is
$$ f''(x) = \frac{2\ln(x) - 3}{x^3} $$
Steps to Solve
- 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:
$$ f'(x) = \frac{\left(\frac{1}{x}\right)x - \ln(x)(1)}{x^2} = \frac{1 - \ln(x)}{x^2} $$
- 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:
$$ f''(x) = \frac{u'v - uv'}{v^2} = \frac{\left(-\frac{1}{x}\right)x^2 - (1 - \ln(x))(2x)}{(x^2)^2} $$
- 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:
$$ f''(x) = \frac{2\ln(x) - 3}{x^3} $$
The second derivative of the function ( \frac{\ln(x)}{x} ) is
$$ f''(x) = \frac{2\ln(x) - 3}{x^3} $$
More Information
The second derivative provides information about the concavity of the function. If ( f''(x) > 0 ), the function is concave up, and if ( f''(x) < 0 ), it is concave down. The point where ( f''(x) = 0 ) usually indicates inflection points.
Tips
- Forgetting to apply the quotient rule correctly when differentiating the first derivative.
- Failing to simplify the final answer thoroughly, leaving it in a more complex form than necessary.