# Thread: first derivative with natural log

1. ## 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)

2. 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,

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

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

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

now can you continue?

3. 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)

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

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

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

5. 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).

6. 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) ???

7. 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.

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

9. 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.

10. o alright.. thank you