derivative of x^x^2

[Guest]

the problem - find the value of f'(1) when f(x)= x^x^2 (x to the power x to the power 2)

i made it into lnx^x^2

then into (x^2)(lnx)

now i am not sure what to do, im guessing i did a wrong step.

 

[pka]

Use implicit differentiation on \(\displaystyle \ln (y) = x^2 \ln \left( x \right).\)

 

[galactus]

Is that \(\displaystyle (x^{x})^{2}\) or \(\displaystyle (x)^{x^{2}}\)?.


For either, you can use logarithmic differnetiation.

If it's the first one:

\(\displaystyle y=(x^{x})^{2}=x^{2x}\)

Log of both sides:

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

Property of logs:

\(\displaystyle ln(y)=2xln(x)\)

Differentiate:

\(\displaystyle \frac{1}{y}y'=2ln(x)+2\)

Remember, \(\displaystyle y=x^{2x}\):

\(\displaystyle \L\\y'=2(ln(x)+1)x^{2x}\)

If it's the other, you give it a go. Use the same technique.