Answer
$$ h'=t^tt^{t^t}\left(\frac{1}{t}+(1+\ln t )\ln t\right).$$
Work Step by Step
To find the derivative, we have
$$\ln h= \ln t^{t^t}=t^t\ln t\Longrightarrow \frac{h'}{h}=t^t \frac{1}{t}+(t^t)'\ln t.$$
Now, to find the derivative of $ g=t^t $, we get
$$\ln g=t\ln t\Longrightarrow \frac{g'}{g}=\ln t + 1\Longrightarrow g'=t^t(1+\ln t ).$$
Finally, we have
$$ h'=t^{t^t}\left(t^t \frac{1}{t}+t^t(1+\ln t )\ln t\right)=t^tt^{t^t}\left(\frac{1}{t}+(1+\ln t )\ln t\right).$$
