Question

Log differentiation:
Compute the following derivatives. Use log differentiation where appropriate.
d/dx[(lnx)^(x^2)]

Answer 1

original function:
\[d/dx(lnx)^{x^2}\]
Add ln on both sides
\[y=(lnx)^{x^2}=>\ln(y)=\ln((lnx)^{x^2})\]
use the log prop ln*x^r)=rln(x) on the right side:
\[=>\ln(y)=\ln(x^2\ln(x))\]
Take derivative using implicit diff. and remove the y in the denom
\[=>y'/y=y(x) \times (\frac{ 1 }{ x^2\ln(x) })[(2x)(\ln(x))+(x^2)(\frac{1}{x})]\]
Plug in y(x):
\[y' = (lnx)^{x^2}[(\frac{1}{x^2\ln(x)})[(2x)(\ln(x)+(x^2)(\frac{1}{x})]\]

Answer 2

The step with the log property is a bit off:
\[\ln f(x)^{(g(x)}=g(x)\ln f(x)\]
so
\[\ln\left(\ln x\right)^{x^2}=x^2\ln(\ln x)\]

Answer 3

@SithsAndGiggles
I see so making those adjustments my new answer is:
\[\ln(y)=(x^2)(\ln(\ln(x)))\]
\[=>y'/y=y(x) \times(2x)(\ln(\ln(x))+\frac{1}{\ln(x)}* (\frac{1}{x})*(x^2)\]
\[=>y'=[\ln(x)^{x^2}][(2x)(\ln(\ln(x))+(1/\ln(x)\times 1/x)\times x^2]\]

Answer 4

Look good to me, just simplify 1/x*x^2 at the last term. :)

Answer 5

ok, thanks.