Question

How do you find the first and second derivative of `ln[lnx^2+1)]`?

Answer 1

`d/dx ln \ ln(x^2+1) \ = (2x)/((x^2+1)ln(x^2+1))`

`d^2/(dx^2) ln \ ln(x^2+1) = ( 2((1-x^2)ln(x^2+1) - 2x^2)) / ((x^2+1)^2ln^2(x^2+1))`

Explanation:

Suppose we let:

`y= ln \ ln(x^2+1)`

Using the chain rule we have:

`dy/dx = 1/(ln(x^2+1)) * d/dx(ln(x^2+1))`
`" " = 1/(ln(x^2+1)) * 1/(x^2+1) * d/dx(x^2+1)`
`" " = 1/(ln(x^2+1)) * 1/(x^2+1) * 2x`
`" " = (2x)/((x^2+1)ln(x^2+1))`

To find the second derivative we will need to apply the quotient rule:

`(d^2y)/(dx^2) = ( ((x^2+1)ln(x^2+1))(d/dx 2x) - (2x)(d/dx (x^2+1)ln(x^2+1)) ) / ((x^2+1)ln(x^2+1))^2`

`" " = ( 2(x^2+1)ln(x^2+1) - (2x)(d/dx (x^2+1)ln(x^2+1)) ) / ((x^2+1)^2ln^2(x^2+1))`

And using the product rule, combined with the chain rulewe have:

`d/dx (x^2+1)ln(x^2+1) = (x^2+1)(d/dx ln(x^2+1)) + (d/dx (x^2+1))(ln(x^2+1))`
`" " = (x^2+1)(1/(x^2+1)d/dx (x^2+1)) + (2x)(ln(x^2+1))`
`" " = 2x + 2xln(x^2+1)`

Injecting this result into our earlier findings we get:

`(d^2y)/(dx^2) = ( 2(x^2+1)ln(x^2+1) - (2x)(2x + 2xln(x^2+1)) ) / ((x^2+1)^2ln^2(x^2+1))`

`" " = ( 2{(x^2+1)ln(x^2+1) - x(2x + 2xln(x^2+1))} ) / ((x^2+1)^2ln^2(x^2+1))`

`" " = ( 2{(x^2+1)ln(x^2+1) - 2x^2 - 2x^2ln(x^2+1)}) / ((x^2+1)^2ln^2(x^2+1))`

`" " = ( 2{(x^2+1-2x^2)ln(x^2+1) - 2x^2}) / ((x^2+1)^2ln^2(x^2+1))`

`" " = ( 2((1-x^2)ln(x^2+1) - 2x^2)) / ((x^2+1)^2ln^2(x^2+1))`