Question

need help finding derivative of f(t)= t^2/3 log subscript 9 (square root of t + 5 )

Answer 1

\[f(t)=t^{\frac{2}{3}}\log_9(\sqrt{t+5})\]

Answer 2

yes perfect !

Answer 3

chain rule and product rule as well as of course the power rule

Answer 4

im lost

Answer 5

ok lets go slow

\[(fg)'=f'g+g'f\] right?

Answer 6

where to begin

Answer 7

put
\[f(t)=t^{\frac{2}{3}},f'(t)=\frac{2}{3}t^{-\frac{1}{3}}\] that part is easy yes?

Answer 8

separate them and take derivative of first part. which is fairly eadsy

Answer 9

but the log is scary

Answer 10

the hard part is
\[g(t)=\log_9(\sqrt{t+5})\]

Answer 11

your in calculus...learn to love logs now...they're your best friend

Answer 12

so first off
\[\frac{d}{dt}\log_b(t)=\frac{1}{\ln(b)x}\]

Answer 13

but you don't have
\[\log_9(t)\] you have
\[\log_9(\sqrt{t+5})\] so for this you need the chain rule

Answer 14

therefore
\[\frac{d}{dx}\log_9(\sqrt{x+5})=\frac{1}{\ln(9)\sqrt{x+5}})\times \frac{d}{dx}\sqrt{x+5}\]

Answer 15

so is it \[\log _{9}(5+t)^{1/2} = (1)/(\ln(5+t)^{1/2}\]

Answer 16

oops i did it wrong then. sorry. i tried to follow ur formula

Answer 17

not quite

Answer 18

hold on lets make it easy ok?

Answer 19

ok

Answer 20

\[\log_9(\sqrt{x+5})=\frac{1}{2}\log_9(x+5)\] by that simple property of the log that exponents come out front as coefficients

Answer 21

OHHHHH

Answer 22

i hope this step is clear, i am using
\[\log(x^n)=n\log(x)\]

Answer 23

so i was on the right track. so is this the final derivative ?

Answer 24

yes that makes more sense :)

Answer 25

sorry but my time is running out... :-S

Answer 26

this should make life much simpler. and in fact we should have started right away with
\[f(t)=t^{\frac{2}{3}}\log_9(\sqrt{t+5})=\frac{1}{2}t^{\frac{3}{2}}\log_9(x+5)\]\]

Answer 27

so is this the final asnwer ?

Answer 28

or u just rewrote the original problem

Answer 29

no i wrote the original problem. we go right to the answer now

Answer 30

oh okay whew.

Answer 31

\[\frac{1}{2}[\frac{2}{3}t^{-\frac{1}{3}}\log_9(x+5)+t^{\frac{3}{2}}\times \frac{1}{\ln(9)(x+5)}]\]

Answer 32

pulled out the one half right at the beginning, then used the product rule

Answer 33

so iam going to submit it. omg i hope its correct. thanks soooooooo much for putting up with me. i really appreciate it. you are a life saver

Answer 34

its a really long asnwer.

Answer 35

\[=[\frac{1}{3}t^{-\frac{1}{3}}\log_9(x+5)+t^{\frac{3}{2}}\times \frac{1}{2\ln(9)(x+5)}]\]

Answer 36

either way

Answer 37

oh distribute the 1/3 through? that would be more complicated. okay i will stick with wat u have

Answer 38

no no

Answer 39

i just distributed the 1/2

Answer 40

it is along answer because it is a product rule problem

Answer 41

you gotta type this in good luck!

Answer 42

ohhhhhhh okay. omg i mso nervous. i hope its right. thanks a lott through i really appreciate it .

Answer 43

urgh got it wrong. :(