Logarithmic Differentiation: Finding derivative

y = (x^{3}√(x^{2 }+ 1))/(x - 1)

What I did:

I used natural log on both sides.

I brought the exponents into the coefficient.

I also brought the (x - 1) up by doing it to the negative one exponent

Then I used the log properties.

So tell me if this is the correct way:

ln y = 3lnx + (1/2)ln(x^{2 }+ 1) - ln(x - 1)

Then I find the derivative

1/y(dy/dx) = 3/x + (1/2)(1/(x^{2 }+ 1)) - (1/(x - 1))

After that I multiplied both sides by y, then plugged in y, which is the original equation.

So would this be right?

Thanks :)

Re: Logarithmic Differentiation: Finding derivative

You made a small mistake in the derivative of ln(x^{2}+1)^{1/2} but other than that it is correct

Re: Logarithmic Differentiation: Finding derivative

Quote:

Originally Posted by

**Shakarri** You made a small mistake in the derivative of ln(x^{2}+1)^{1/2} but other than that it is correct

Hmm, so first I would put the exponent into the coefficient right?

(1/2)ln(x^{2}+1)

Um, would I then have the log go into the x and 1?

Yeah sorry, doing the derivative of 1 variable for ln is easy for me.

So if it was that (1/2)ln(x^{2}+1)

I wouldn't just do what's in the parenthesis and put it all down under 1?

Then multiply it by 1/2?

So like 1/(2(x^{2}+1))?

Re: Logarithmic Differentiation: Finding derivative

$\displaystyle v= ln(x^2+1)$

let $\displaystyle u=(x^2+1)$

$\displaystyle \frac{du}{dx}=2x$

$\displaystyle v=ln(u)$

$\displaystyle \frac{dv}{du}=\frac{1}{u}$

$\displaystyle \frac{dv}{dx}= \frac{du}{dx}\cdot \frac{dv}{du} $

$\displaystyle \frac{dv}{dx}=2x \cdot \frac{1}{u}= \frac{2x}{(x^2+1)}$

Re: Logarithmic Differentiation: Finding derivative

Quote:

Originally Posted by

**Shakarri** $\displaystyle v= ln(x^2+1)$

let $\displaystyle u=(x^2+1)$

$\displaystyle \frac{du}{dx}=2x$

$\displaystyle v=ln(u)$

$\displaystyle \frac{dv}{du}=\frac{1}{u}$

$\displaystyle \frac{dv}{dx}= \frac{du}{dx}\cdot \frac{dv}{du} $

$\displaystyle \frac{dv}{dx}=2x \cdot \frac{1}{u}= \frac{2x}{(x^2+1)}$

Oh right, now I remember!

So it's the derivative of that over the original!

Thanks!

Then after that you would multiply it by 1/2 right?

Which will be 2x/(2(x^{2}+1))