Question

derivative of y=x^(3/2)log(base3)squareroot(x+7)

Answer 1

\(\large\rm y=x^{3/2}\cdot\log_3\sqrt{x+7}\)
This guy?
Hmm this looks like a fun one :) lol

Answer 2

\[y= x ^{3/2}\log_{3}\sqrt{x+7}\]

Answer 3

yeah lol

Answer 4

Do you remember your change of base formula for logs?\[\large\rm \log_{a}(b)=\frac{\ln(b)}{\ln(a)}\]We would choose natural log like this ^
Because it's easier to differentiate.

Answer 5

Unless you have a special formula memorized for log derivatives :)
\[\large\rm y=x^{3/2}\cdot\frac{\ln\sqrt{x+7}}{\ln3}\]

Answer 6

so it would be \[(3/2)x ^{1/2}*(\ln \sqrt{x+7}/\ln3)\]

Answer 7

?

Answer 8

No no no :O
Product rule.

Answer 9

what should i do then?

Answer 10

I'll do the "set up" and you can let me know if it makes sense:\[\large\rm y'=\color{royalblue}{\left(x^{3/2}\right)'}\cdot\frac{\ln\sqrt{x+7}}{\ln3}+x^{3/2}\cdot\color{royalblue}{\left(\frac{\ln\sqrt{x+7}}{\ln3}\right)'}\]This is our product rule "set up".
I haven't taken any derivatives yet.
I'm just setting it up.
We need to differentiate the blue parts.

Answer 11

is the first blue part \[(3/2)x ^{(3/2)-1}\]

Answer 12

\[\large\rm y'=\color{orangered}{\left(\frac{3}{2}x^{1/2}\right)}\cdot\frac{\ln\sqrt{x+7}}{\ln3}+x^{3/2}\cdot\color{royalblue}{\left(\frac{\ln\sqrt{x+7}}{\ln3}\right)'}\]Ok ya that sounds right!

Answer 13

This other guy is a bit tricky though, ya? :)

Answer 14

\[1/((1/2)x ^{-1/2}+(1/2)7^{-1/2}) /(1/3)\]

Answer 15

?

Answer 16

No.
And oh boy, did you ever make things complicated :D lol

Answer 17

Let's back up a sec, apply a helpful log rule before we take the derivative.

Answer 18

sorry lol

Answer 19

\[\ln \sqrt{x+7}-\ln 3\]

Answer 20

?

Answer 21

First notice that ln3 is just a constant, so we can pull it out of the differentiation process and ignore it.\[\large\rm \color{royalblue}{\left(\frac{\ln\sqrt{x+7}}{\ln3}\right)'}=\frac{1}{\ln3}\color{royalblue}{\left(\ln\sqrt{x+7}\right)'}\]

Answer 22

what about the x+7

Answer 23

Next rewrite your square root as a 1/2 power,\[\large\rm \frac{1}{\ln3}\color{royalblue}{\left(\ln(x+7)^{1/2}\right)'}\]And apply another log rule,\[\large\rm =\frac{1}{\ln3}\color{royalblue}{\left(\frac{1}{2}\ln(x+7)\right)'}\]And again pull the 1/2 out front because it's just a constant,\[\large\rm =\frac{1}{2\ln3}\color{royalblue}{\left(\ln(x+7)\right)'}\]The problem becomes a lot easier to deal with at this point.

Answer 24

This is one of the things that can make Calc a difficult subject, remember all of these silly Algebra rules :) They're really helpful though.

Answer 25

does the x+7 become 8

Answer 26

or does it become 2

Answer 27

I don't understand what you're asking

Answer 28

does \[(\ln(x+7))\prime = \ln(2)\prime = 1/2\]

Answer 29

?? 0_o

Answer 30

I have no idea where that 2 is coming from.. this is so confusing

Answer 31

Do you remember your derivative for ln(x) ?

Answer 32

isnt the derivative of a constant = 1?

Answer 33

so the derivative of x = 1 and the derivative of 7 = 1

Answer 34

so 1+1 = 2

Answer 35

???

Answer 36

The derivative of a constant is = 0.
x is not a constant.

But let's not worry about that. That's not what we're dealing with here.
I'm asking if you know the derivative of \(\large\rm \ln(x)\)

Answer 37

is it 1?

Answer 38

... No :(

Answer 39

isnt the derivative of ln(x) = 1/x

Answer 40

Yes.

Answer 41

So then by that same rule,\[\large\rm \frac{d}{dx}\ln(x+7)=\frac{1}{(x+7)}\]

Answer 42

wait so is the answer \[(3/2)x ^{1/2}-((\ln(\sqrt{x+7})/\ln3)*x ^{3/2})*(1/x+7)\]

Answer 43

No :(

Answer 44

what is the right answer?

Answer 45

By the steps I was showing you before,
the derivative of that second blue thing we were working on,
would be,\[\large\rm \color{royalblue}{\left(\frac{\ln\sqrt{x+7}}{\ln3}\right)'}=\color{orangered}{\frac{1}{2\ln3(x+7)}}\]

Answer 46

\[\large\rm y'=\color{orangered}{\left(\frac{3}{2}x^{1/2}\right)}\cdot\frac{\ln\sqrt{x+7}}{\ln3}+x^{3/2}\cdot\color{royalblue}{\left(\frac{\ln\sqrt{x+7}}{\ln3}\right)'}\]
So I guess our final result is something like this,\[\large\rm y'=\color{orangered}{\left(\frac{3}{2}x^{1/2}\right)}\cdot\frac{\ln\sqrt{x+7}}{\ln3}+x^{3/2}\cdot\color{orangered}{\frac{1}{(2\ln3)(x+7)}}\]

Answer 47

Practice more problems cp :O
Really really need to study!

Answer 48

OH OKAY

Answer 49

I GET IT NOW

Answer 50

THANK YOU SO MUCH