What is y'' for y= (ln(5x))/x^3?

We have y = ln 5x / x^3. We have to determine
y''.

y = ln 5x / x^3 = ln 5x *
x^-3

Use the product rule

y' =
[ln 5x]' * x^-3 + ln 5x *[x^-3]'

=> y' = (5 / 5x)*
x^-3 -3*ln 5x * x^-4

=> y' = x^-4 - 3*ln 5x *
x^-4

=> x^-4( 1 - 3* ln
5x)

y'' = [x^-4]'( 1 - 3* ln 5x) + (x^-4)[1 - 3* ln
5x]'

=> y'' = -4*x^-5 * ( 1 - 3* ln 5x) + (x^-4)[-
3*5/5x]

=> y'' = -4*x^-5 * ( 1 - 3* ln 5x)
-3*(x^-5)

=> y'' = x^-5 ( -4 + 12 ln 5x -
3)

=> y'' = x^-5(12*ln 5x -
7)

The required result is y'' = (12*ln 5x -
7)/ x^5