Second Derivative

Question

Find the second derivative of: ln(x^5 + x)

What I got as an answer was -5x^8-5x^4+20x^3/(x^5+1)^2, but when I check with other websites to confirm my answer they provide me with a different answer, and I noticed that for the first derivative they factor the x in the denominator which is something I didn't do, and I am not sure if I'm incorrect.

Answer 1

Hi Dwayn,

Thanks for your question!

For the first derivative, I get

dy/dx = d/dx( ln(x^5 + x) )

= 1/(x^5 + x) * d/dx(x^5 + x)

= (5x^4 + 1)/(x^5 + x).

I'll leave the denominator unfactored, to be consistent with how you've done the problem. Then, for the second derivative, I get

d^2y/dx^2 = d/dx[ (5x^4 + 1)/(x^5 + x) ]

= [(x^5 + x) * d/dx(5x^4 + 1) - (5x^4 +1) * d/dx(x^5 + x)]/[(x^5 + x)^2]

= [(x^5 + x) * (20x^3) - (5x^4 + 1)*(5x^4 + 1)]/[(x^5 + x)^2]

= [20x^8 + 20x^4 - (25x^8 + 10x^4 + 1 )]/[(x^5 + x)^2]

= (-5x^8 + 10x^4 - 1)/(x^5 + x)^2.