Question

derivative of f(x)=ln((x^(8)-8)^(5)) ??

Answer 1

\[\large f(x)=\ln\left[(x^8-8)^5\right]\]Can you see the formatted text above?
So we need to take the derivative of this, yes? :)

Answer 2

Yes! thats it!

Answer 3

Do you recall the derivative of \(\large \ln x\) ?

Answer 4

1/x?

Answer 5

Oh I guess we could apply a rule of logarithms to make this a tad easier.\[\large \log (a^b)=b \cdot \log a\]

So before we do anything, let's rewrite our equation like this,\[\large f(x)=5\cdot\ln\left[(x^8-8)\right]\]Using the above mentioned rule to bring the 5 out front.

Answer 6

Yes, very good, 1/x.
We'll be using that derivative rule for our first step.

Answer 7

okay sounds good

Answer 8

\[\large f(x)=5\cdot\ln\left(x^8-8\right)\]
\[\large f'(x)=5\cdot \frac{1}{x^8-8}\color{royalblue}{\left(x^8-8\right)'}\]

Ignore the blue term for now, and ignore the 5. The 5 will just follow us along for the ride.
Understand the first step?
Taking the derivative of the log, places EVERYTHING inside the log, into the denominator.

Answer 9

okay that makes sense!

Answer 10

The blue term shows up due to the chain rule.
The chain rule tells us we have to multiply by the derivative of the inner function.

The little prime on the outside is to show that we need to take it's derivative still.
What do you get taking the derivative of the blue part?

Answer 11

\[8x ^{7}\]

Answer 12

?

Answer 13

\[\large f'(x)=5\cdot \frac{1}{x^8-8}\color{royalblue}{\left(8x^7\right)}\]Yup sounds good :)
To finish up, you could put all that stuff on top of the fraction so it looks a little nicer.\[\large f'(x)=\cdot \frac{5\cdot 8x^7}{x^8-8}=\frac{40x^7}{x^8-8}\]

Answer 14

can anything multiplied by a fraction be moved to the numerator like that?

Answer 15

Yes, here are a couple things to remember.\[\large 5 = \frac{5}{1}\]
So when we're multiplying this,\[\large 5\cdot \frac{1}{x^8-8}\]
We're really just multiplying fractions,\[\large \frac{5}{1}\cdot \frac{1}{x^8-8}\]

Answer 16

And then from there, you just need to remember how to multiply fractions (top by top, bottom by bottom).

Answer 17

great! so is 40x^7 / 8x^8 the final answer?

Answer 18

yup! good job.

Answer 19

I meant: x^8-8

Answer 20

awesome thanks!!1

Answer 21

hehe that' what i thought u meant XD

Answer 22

thank you!!