# first derivative with natural log

• Oct 9th 2008, 03:25 PM
Cucc1137
first derivative with natural log
Hi, I just spent a while doing math homework and I am really stuck on this last question. It reads find the first derivative of

y = x^ln(x)

• Oct 9th 2008, 03:27 PM
Jhevon
Quote:

Originally Posted by Cucc1137
Hi, I just spent a while doing math homework and I am really stuck on this last question. It reads find the first derivative of

y = x^ln(x)

you have to do logarithmic differentiation here. so begin by logging both sides. we get,

$\displaystyle \ln y = \ln x^{\ln x}$

$\displaystyle \Rightarrow \ln y = \ln x \ln x$

$\displaystyle \Rightarrow \ln y = (\ln x)^2$

now can you continue?
• Oct 9th 2008, 03:29 PM
mr fantastic
Quote:

Originally Posted by Cucc1137
Hi, I just spent a while doing math homework and I am really stuck on this last question. It reads find the first derivative of

y = x^ln(x)

$\displaystyle \ln y = \ln x \ln x = (\ln x)^2$

Differentiate both sides with respect to x (use implicit differentiation on the left hand side):

$\displaystyle \frac{1}{y} \frac{dy}{dx} = 2 \, (\ln x) \, \left( \frac{1}{x}\right)$.

Now make $\displaystyle \frac{dy}{dx}$ the subject and substitute $\displaystyle y = x^{\ln x}$.
• Oct 9th 2008, 03:30 PM
Cucc1137
is that last step the derivative or do you take the derivative of that part?
• Oct 9th 2008, 03:33 PM
mr fantastic
Quote:

Originally Posted by Cucc1137
is that last step the derivative or do you take the derivative of that part?

I have no idea what you mean.

I gave you that

$\displaystyle \frac{1}{y} {\color{red}\frac{dy}{dx}} = 2 \, (\ln x) \, \left( \frac{1}{x}\right)$.

Now make the stuff in red the subject (by multiplying both sides by y).
• Oct 9th 2008, 03:34 PM
Cucc1137
so is it

dy/dx = (2)(lnx)(1/x)(x^lnx)

can u simplify that down alot of just the basic (2lnx/x)(x^lnx) ???
• Oct 9th 2008, 03:37 PM
mr fantastic
Quote:

Originally Posted by Cucc1137
so is it

dy/dx = (2)(lnx)(1/x)(x^lnx)

can u simplify that down alot of just the basic (2lnx/x)(x^lnx) ???

Yes and yes.
• Oct 9th 2008, 03:38 PM
Cucc1137
so what i have at the end is right .. or can you simplify more??
• Oct 9th 2008, 03:40 PM
mr fantastic
Quote:

Originally Posted by Cucc1137
so what i have at the end is right Mr F says: Yes. .. or can you simplify more?? Mr F says: No.

If it could be simplified further I would have suggested that to you.
• Oct 9th 2008, 03:43 PM
Cucc1137
o alright.. thank you