Question

Find derivative of f(x)=ln(sqrt((4x+5)/(3x-9)))

Answer 1

\[\ln \sqrt{(4x+5)/(3x-9)}\]

Answer 2

Come on, give it a try! :-)

(d/dx)ln(x) = 1/x

(d/dx)sqrt(x) = 1/[2sqrt(x)]

(d/dx)(f(x)/g(x)) = [g(x)f'(x) - f(x)g'(x)]/[g(x)^2]

And the Chain Rule!!

That's all you need. An excellent survey problem.

Answer 3

i actually got to a point where i plugged in x into the ln derivative, and did the derivative of the division part, now i think i am stuck with the math part

Answer 4

It's nice and sequential\[\frac{d}{dx}\ln\left(\sqrt{(4x+5)/(3x-9)}\right)\]
\[\frac{1}{\sqrt{(4x+5)/(3x-9)}}\cdot\frac{1}{2\cdot\sqrt{(4x+5)/(3x-9)}}\cdot\frac{(3x-9)(4)-(4x+5)(3)}{(3x-9)^{2}}\] The logarithm, the square root, the quotient - each connected by the Chain Rule. It ain't pretty, but it shouldn't be scary.

Answer 5

thats what i am on! so all i do is multiply it out? the roots will disappear? the the derivative of the quotient will become -51/(3x-9)^2

Answer 6

No, the square roots do not disappear. \[\sqrt{x}\sqrt{x} = x \ne 1\] You should get \[-\frac{17}{2(x-3)(4x+5)}\] Do the algebra, don't guess.

Answer 7

so i am multiplying out (4x+5/3x-9) twice? is my approach right?

Answer 8

Not at all, since you seem to be just guessing. \[\sqrt{a/b}\cdot\sqrt{a/b}= a/b\] Do the algebra. Don't worry if your eyes don't see it. Let the notation help you.

Answer 9

\[(1/(2((4x+5)/(3x-9)))(-51/(3x-9)^2) \]

Answer 10

THAT, My Friend, was not guessing!

Now simplify some more. There's lots of (3x-9) still floating around. You can rewrite the first part if you like \[\frac{1}{2\frac{4x+5}{3x-9}} = \frac{3x-9}{2(4x+5)}\] It may be more clear like that.

Answer 11

I never really guessed, I had the parts down but the algebra was confusing me. So now i have \[-51\div(2\times(4x +5)(x-3))\] I was wondering how did you divide 3 from only one part of the equation to get 17, but didnt diviade the other side also

Answer 12

Common factor of 3 in (3x-9). Factor it out. There aren't teo sides. It's just one expression.