Question

f(x)=x^6x
find f'(x) by using antilogarithm differentiation

Answer 1

logarithmic not antilogarithm.. missed typed

Answer 2

you have a choice, although the work is identical
you can write
\[x^{6x}=e^{x\ln(6x)}\] and then take the derivative using the chain rule,
or you can take the log, get
\[x\ln(6x)\] then take the derivative of that, and once done multiply by the original function

Answer 3

all the work is the same however, you have to find the derivative of
\[x\ln(6x)\] no matter which method you choose

Answer 4

Where does the x infront of ln(6x) come from? and why is that there? @satellite73

Answer 5

so it would be x (6/6x)

Answer 6

?

Answer 7

@Jhannybean what is the definition of \(x^{6x}\) or for that matter how would you define
\[4^{\sqrt2}\]

Answer 8

no

Answer 9

for \(x\ln(6x)\) you need the product rule

Answer 10

\[(fg)'=f'g+g'f\] with
\[f(x)=x, f'(x)=1, g(x)=\ln(6x), g'(x)=\frac{1}{6}\]

Answer 11

oops i meant \(g'(x)=\frac{1}{x}\)

Answer 12

@Jhannybean i see i made a mistake, good catch

Answer 13

ln y = 6x lnx

\[\huge y = e^{6x \ln x}\]

Answer 14

\[x^{6x}=e^{6x\ln(x)}\]

Answer 15

Yeah, the 6 disappeared.

Answer 16

yeah it was wrong, you are right
also @agent0smith caught me too

Answer 17

Great explanation you guys! :D

Answer 18

can't get away with anything here
ok since i messed up, lets do it right
you need the derivative of \(6x\ln(x)\) which still requires the product rule
this time is is \[6\ln(x)+6x\times \frac{1}{x}=6\ln(x)+6\]

Answer 19

i got it Thanks guys

Answer 20

how do u find the derivated for 4^(x)log[7](x)
@satellite73

Answer 21

7 is the base of log

Answer 22

the derivative of \(b^x\) is \(b^x\times \ln(b)\)

Answer 23

and
\[\log_b(x)=\frac{\ln(x)}{\ln(b)}\] by the all powerful change of base formula (which says all logs are the same) and so
\[\frac{d}{dx}[\log_b(x)]=\frac{1}{x\ln(b)}\]

Answer 24

i got the derivative, but do i need to use the product rule or not?

Answer 25

If you got the derivative why do you still need to use the product rule?

Answer 26

so would the answer be 4^xln(4)(1/xln(7)) or 4^xln(4)(log_7(x))+4^(x)(1/xln(7))

Answer 27

nvm.. Thanks for yall's help

Answer 28

No you're overcomplicating it :)

Answer 29

lol