<s>[INST] Finding the derivative of $x$ to the power something that is a function of $x$ if $y = x^{(x+1)^\frac12}$
then how can I get the first derivative of $y$?
 [/INST] Another way to solve it is to take $\ln$ of both sides, and apply implicit differentiation:
$$y=x^{(x+1)^{\frac {1}{2}}}$$
$$\ln y=\ln x^{(x+1)^{\frac {1}{2}}}$$
Rewriting using $\log$ properties:
$$\ln y=(x+1)^{\frac {1}{2}} \cdot \ln x$$
Now take the derivative implicitly:
$$\frac{1}{y}\cdot y' = \frac{1}{2}(x+1)^{-\frac{1}{2}} \ln x + (x+1)^{\frac {1}{2}} \cdot \frac{1}{x}$$
We want $y'$, so we multiply $y$ on both sides to isolate $y'$.
$$ y' = y \cdot \left( \frac{1}{2}(x+1)^{-\frac{1}{2}} \ln x + (x+1)^{\frac {1}{2}} \cdot \frac{1}{x} \right)$$
We want the right hand side to be in terms of $x$. The original problem says $y=x^{(x+1)^{\frac {1}{2}}}$, so
$$ y' = \big( x^{(x+1)^{\frac {1}{2}}} \big) \cdot \left( \frac{1}{2}(x+1)^{-\frac{1}{2}} \ln x + (x+1)^{\frac {1}{2}} \cdot \frac{1}{x} \right)$$
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