first derivative with natural log

**Cucc1137** · October 9th 2008, 03:25 PM

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)

can someone please help me so i can move on.. Thanks!!!

**Jhevon** · October 9th 2008, 03:27 PM

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)

can someone please help me so i can move on.. Thanks!!!

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

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

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

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

now can you continue?

**mr fantastic** · October 9th 2008, 03:29 PM

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)

can someone please help me so i can move on.. Thanks!!!

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

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

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

Now make $\frac{dy}{dx}$ the subject and substitute $y = x^{\ln x}$ .

**Cucc1137** · October 9th 2008, 03:30 PM

is that last step the derivative or do you take the derivative of that part?

**mr fantastic** · October 9th 2008, 03:33 PM

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

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

Now make the stuff in red the subject (by multiplying both sides by y).

**Cucc1137** · October 9th 2008, 03:34 PM

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) ???

**mr fantastic** · October 9th 2008, 03:37 PM

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.

**Cucc1137** · October 9th 2008, 03:38 PM

so what i have at the end is right .. or can you simplify more??

**mr fantastic** · October 9th 2008, 03:40 PM

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.

**Cucc1137** · October 9th 2008, 03:43 PM

o alright.. thank you

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