Question

can someone help me step by step?

find the derivative

y=ln((e^(7x))/(sqrt(4x-5)))

Answer 1

before you begin taking the derivative, use the properties of the log to make this expression easier to differentiate

Answer 2

it is the log of the whole thing right?

Answer 3

yes.

Answer 4

so first step would be, before beginning to take the derivative, rewrite as
\[\ln(e^{7x})-\frac{1}{2}\ln(4x-5)\]

Answer 5

are those steps clear?

Answer 6

why is 1/2 there and not (4x-5)^(1/2)

Answer 7

i used two facts
\[\log(\frac{a}{b})=\log(a)-\log(b)\] and
\[\log(a^n)=n\log(a)\]

Answer 8

because
\[\log(\sqrt{4x-5})=\log((4x-5)^{\frac{1}{2}})=\frac{1}{2}\log(4x-5)\]

Answer 9

but 1/2 is outside the () and not inside ?

Answer 10

You can use the rule like that. As long as it's insside the log.

Answer 11

the one half comes right out front as a multiplier

Answer 12

on other words, \(\log(\sqrt{x})=\frac{1}{2}\log(x)\)

Answer 13

then one more step before differentiating
since log and exp are inverse functions, you have
\[\log(e^{7x})=7x\]

Answer 14

okay, so do i need to use product rule for 1/2ln(4x-5)?

Answer 15

oh no not at all

Answer 16

\(\frac{1}{2}\) is just a constant , leave it there

Answer 17

okay

Answer 18

Well you could... But it's a waste of time.

Answer 19

for example if you wanted the derivative of \(\frac{1}{2}x^3\) you do not use the product rule, you just say \(\frac{3}{2}x^2\)

Answer 20

gotcha. the power rule^

Answer 21

so now you have
\[7x+\frac{1}{2}\ln(4x-5)\] so the only rule you need now is the chain rule for the second part, and also knowing what the derivative of the log is

Answer 22

typo there, i meant
\[7x-\frac{1}{2}\ln(4x-5)\]

Answer 23

okay imma try it/

Answer 24

ok let me know what you get