third derivative

Question

f(x) = ln(ln(7x)) = f'''(x) = ?

Answer 1

Good catch, Ragmar.  The answer has been updated.

 

 

f'(x) = 1/ln(7x) d(ln(7x))/dx

 

= 1/ln(7x) 1/7x d(7x)/dx

 

= 1/ln(7x) (1/7x) (7)

 

= 1/[x ln(7x)]

 

 

f"(x) = (-1/x²)(1/ln(7x)) + (1/x)(-1/ln²(7x))(1/x)

 

= -1/[x²ln(7x)] - 1/[x² ln²(7x)]

 

 

= -[ln(7x) + 1] / [x² ln²(7x)]

 

 

f"'(x) = {-[x² ln²(7x)][1/x]+[ln(7x) + 1](2x ln²(7x) + 2x ln(7x)} / [x⁴ ln⁴(7x)]

 

 = [-x ln²(7x) + 2xln³(7x) + 2xln²(7x) + 2xln²(7x) + 2x ln(7x)] / [x⁴ ln⁴(7x)]

 

= [2 + 3 ln(7x) + 2 ln²(7x)] / [x³ ln³(7x)]