$[8]$ Fun with the FTC$!$

In the following, $F$ just represents an abstract function, not an antiderivative.

$($a$)$ If $f(x)=sin(x^2),$ then what are $f(2),f(t),f(t+1),f(t^2),$ and $f(x^2)?$

$($b$)$ If $f(x)=sin(x^2),$ then what is $f^{\text{'}}(x)?$

$($c$)$ If $g(x)=F(x^2),$ then what are $g(4),g(\sqrt[\phantom{2}]{t}),$ and $g(x+1)?($Your answers will be expressed in terms of $F.)$

$($d$)$ If $g(x)=F(x^2),$ then what is $g^{\text{'}}(x)?($Again$,$ your answer will be expressed in terms of $F.)$

$($e$)$ If $g(x)=F(x^2),$ then what are $g^{\text{'}}(4)$ and $g^{\text{'}}(\sqrt[\phantom{2}]{t})?$

$($f$)$ If $G(x)={\displaystyle \underset{0}{\overset{x}{}}}cos{t}^{2}dt,$ what is $G^{\text{'}}(x)?$

$($Hint: Do not try to integrate $cos{t}^2.$ Use the FTC$.$ Also, note that $cos{t}^2$ means $cos(t^2),$ not... anything else.$)$

$($g$)$ If $h(x)={\displaystyle \underset{0}{\overset{\sqrt[\phantom{2}]{x}}{}}}cos{t}^{2}dt,$ what is $h^{\text{'}}(x)?$

$($You often see problems like this in calculus textbooks in the chapter on the Fundamental Theorem of Calculus. I always used to think they were kind of silly, but I have learned since that it can sometimes be useful to know how to differentiate a function defined in this way.$)$