I stumbled upon a sneaky example while reading about the Fundamental Theorem of Calculus here $$\frac d{dx}\int_1^\sqrt xt^tdt$$ and it makes me question my whole understanding.
I know that the FTC says $$\frac d{dx}\int_a^x f(t)dt\,=\, f(x)$$ so I would say that $f(x)=\sqrt x^\sqrt x$ but this is incorrect I should get $$f(x)=\frac 12 x^{\frac {\sqrt x}2-\frac 12}$$ which I see as $$f(x) = \frac 12 \left(\frac 1{\sqrt x}\right)x^{\frac {\sqrt x}2}$$ I am confused, why does this look like the result I get from $\frac d{dx}\sqrt x^\sqrt x$ and not $\sqrt x^\sqrt x$? I can almost get to $\frac 12 x^{\frac {\sqrt x}2-\frac 12}$ when I take the derivative of $\sqrt x^{\sqrt x}$, I let $u=\sqrt x$ such that $$\frac d{dx}x^u\cdot\frac {du}{dx} = \left(\frac {\sqrt x}2 x^{\frac{\sqrt x}2-1}\right)\left(\frac 1{2\sqrt x}\right)$$ $$\frac d{dx} x^{\frac {\sqrt x}2} = \frac 14 x^{\frac {\sqrt x}2-1}$$ I want to make sure I grasp those concepts before I move on and not just blindly apply formulas so I would like to know what is really going on. Thanks in advance for your help, this is greatly appreciated!