Question

Use logarithmic differentiation to find the derivative of the function. y= x^(ln(7x))

Answer 1

log base x (y) = ln(7x)
crap... haha, i can't remember what to do next

Answer 2

\[\Large y=x^{\ln(7x)}\]Taking the natural log of both sides,\[\Large \ln y=\ln [x^{\ln(7x)}]\]

Applying a rule of logs:\[\Large \color{teal}{\log(a^b)\quad=\quad b\cdot \log(a)}\]gives us,\[\Large \ln y=\ln(7x)\ln[x]\]

Answer 3

Confused by any of that? :o

Answer 4

I understand up to that but I got stuck when I tried to continue.

Answer 5

So now we need to take the derivative of each side, with respect to x,\[\Large \color{royalblue}{\left(\ln y\right)'} \quad=\quad \color{royalblue}{\left(\ln7x\right)'}\ln x+\ln7x\color{royalblue}{\left(\ln x\right)'}\]
Applying the product rule to the right side.
So here is our product rule setup.
All of the blue parts represent derivatives we need to take.

Answer 6

alright, so that is:
\[y/y' = \ln x/x + \ln 7x/x\]

right?

Answer 7

mmm yah looks good so far!

Answer 8

woops, left side should be y'/y

Answer 9

oh yea! Thank you :)

Answer 10

got it from there? c:

Answer 11

Oh and \(\Large \color{royalblue}{\text{Welcome to OpenStudy! :)}}\)