Question

Find the second derivative of the function f(x) = log5 x^3.

Answer 1

I know the first derivative is \[\frac{ 3 }{ xln5 }\]

but then i'm stuck...

Answer 2

If I were you I'd start by using the change of base formula to convert the logarithm to a quotient of natural logarithms, which are really easy to derive.
OR Using what you just typed: 3/ln5 is a constant isn't it?

Answer 3

What do you think now, in light of the rules you may have learned?

Answer 4

Well I think I need to use the chain rule... but I don't know for sure... And I have no idea what the derivative of xln5 would be...

Answer 5

Constant multiple rule. What's the derivative of 3/x?

Answer 6

-3/x^2?

Answer 7

Yes, and since ln5 is a constant, you can just insert it into the denominator. Maybe I made this difficult.\[\frac{ d }{ dx }(\frac{ 1 }{ x })=\frac{ -1 }{ x ^{2} }\]\[\frac{ d }{ dx }(\frac{ 3 }{ xln 5})=\frac{ 3 }{ \ln 5 }* \frac{ d }{ dx }(\frac{ 1 }{ x })\]because 3/ln5 is a constant. Check it on a calculator if you like

Answer 8

hmmm... the answer i'm given is \[-\frac{ 3 }{ \ln5 } x^-2\]

Answer 9

Think about it: What does a negative exponent do?

Answer 10

oh pellet... makes it a fraction.

Answer 11

Yep. Btw, are you at all interested in using the change of base formula on future problems?

Answer 12

yes i am interested.

Answer 13

i need to learn it... well i'm supposed to already know it. i'm a little behind in my class.

Answer 14

That's all right. Math is something you can learn any time\[\log_{b}x=\frac{ \ln x }{ \ln b } \] or (treating ln b as a constant)\[=\frac{ 1 }{ \ln b }(\ln x)\] This saves you from learning a rule just for derivatives of logs in other bases. The derivatives of natural logs are really easy.

Answer 15

If you need any more help, just mention me (that @ thing) or message me

Answer 16

okay. thanks a lot for your help today.

Answer 17

yw