Question

Differentiate the function y = 2x*log[10](sqrt(x))

Answer 1

y' = [(2x)/(ln(10)*sqrt(x)] + [2*log[10](sqrt(x))]

Answer 2

correct so far?

Answer 3

or no the derivative of log[10](sqrt(x)) would be (1/[sqrt(x)ln(10)])*(1/[2sqrt(x)])

Answer 4

no it's not

Answer 5

well according to the textbook the answer is pretty different

Answer 6

alright well i've got 1/ln(10) + 2*log[10](sqrt(x))

Answer 7

that's not final though so help me from there

Answer 8

somehow the second term (after the +) needs to equal log[10](x) does that make sense

Answer 9

isn't it something like this
http://www.wolframalpha.com/input/?i=d%2Fdx%28+2x*log+base+10+%28sqrt%28x%29%29%29

Answer 10

sure i guess it's something like that bro jesus you're a great help

Answer 11

but that's still not the textbook answer

Answer 12

let me test,

dy/dx = 2{dx/dx*log[10](sqrt(x))+x*d(log[10](sqrt(x)))/dx}

= 2{log[10](sqrt(x))+x*d(ln(sqrt(x)))/dx*1/ln10}
= 2{log[10](sqrt(x))+x*1/sqrt(x)*1/2*1/sqrt(x)*1/ln10}

= 2{log[10](sqrt(x))+1/2ln10} my answer.

Answer 13

alright here look the textbook answer is 1/ln(10) + log[10](x).

Answer 14

i got almost same answer ... except 1/ln(10) + 2log[10](x)

Answer 15

i got as far as 1/ln(x) + 2(log[10](x)), the latter of which = 2[log(sqrt(x))/log(10)], so [2*log(sqrt(x))] / log(10) = log(sqrt(x))^2/log(10) ?? which = log(x)/log(10)? which = log[10](x) ???? is that correct? does that make sense? i'm reeeal rusty on the log business

Answer 16

nah bro according to the product rule that other end of the + sign is gonna be the derivative of 2x * the log[10](sqrt(x)). unless that's not the case. but i'm pretty sure it is. don't see any way around it really. and you got the derivative of 2x multiplying the other factor which is supposed to be left as is, so.... you tell me..... if and how i'm wrong.....

Answer 17

use this property of log to convert it into natural log.
\[ \log_ax = \frac{\ln x}{\ln a} \]
I think there's a better example at wikipedia.

Answer 18

i know that, and it still doesn't explain how 2*log[10](sqrt(x)) = log[10](x).

Answer 19

oh unless... that equals log[10]([sqrt(x)]^2) ???? is that it?

Answer 20

oh so it is