derivative of logarithmic functions

[fais_attention]

Doing math homework.

f(x)=5x^(ln(x))
Find f ' (6).

I got 25((2ln(6))/6)(6^(ln(6))) when I worked it out myself, but that is apparently wrong. To get that answer, I took the natural log of both sides and then used the product rule of derivatives to get the derivative, then multiplied through by y to get rid of the (1/y)(dy/dx).

So, I tried to work it out again.

I took the natural log of both sides again, only this time i included the coefficient 5. (and replaced f(x) with y)
ln(y)=ln(x)ln(5x)
I differentiated that, and got
dy/dx=(5x^(ln(x)))(ln(5x)/x + ln(x)/x)
So then I plugged 6 into that derivative, and it got confusing. I think I might be making a mistake in the first part, where 5 x 6 is raised to the ln(6). I put in both 30^(ln(6)) and (5)(6^(ln(6)) for that part of the answer.

I would really appreciate it if someone would point me in the right direction.

 

[skeeter]

\(\displaystyle y = 5x^{\ln{x}}\)

\(\displaystyle \ln{y} = \ln(5x^{\ln{x}})\)

\(\displaystyle \ln{y} = \ln{5} + (\ln{x})^2\)

\(\displaystyle \frac{y'}{y} = \frac{2\ln{x}}{x}\)

\(\displaystyle y' = y \cdot \frac{2\ln{x}}{x}\)

\(\displaystyle y' = 5x^{\ln{x}} \cdot \frac{2\ln{x}}{x}\)