Answer
$2at+b$
Work Step by Step
SUMMARY (rules in differential notation):
1. The Power Rule$:\ \ \ \displaystyle \frac{d}{dx}[x^{n}]=n\cdot x^{n-1 } $
2. Sum Rule: $\displaystyle \ \ \ \frac{d}{dx}[f\pm g](x)=\frac{d}{dx}[f(x)]\pm\frac{d}{dx}[g(x)] $
3. Constant Multiple Rule:$\ \ \displaystyle \frac{d}{dx}[cf(x)]=c\cdot\frac{d}{dx}[f(x)] $
4. Constant times x:$\ \ \ \displaystyle \frac{d}{dx}(cx)=c $
5. Constant:$\displaystyle \ \ \ \ \ \frac{d}{dx}(c)=0 $
6. $\displaystyle \frac{d}{dx}(|\mathrm{x}|)=\frac{|x|}{x}$, (from sec.10.6)
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$ \displaystyle \frac{d}{dt}[at^{2}+bt+c)$ = $\ \ \ $...(2)
$=\displaystyle \frac{d}{dt}(at^{2})+\frac{d}{dt}(bt)+\frac{d}{dt}(ct^{0})$ = $\ \ \ $...($3$)
$=a\displaystyle \frac{d}{dt}(t^{2})+b\frac{d}{dt}(t^{1})+c\frac{d}{dt}(t^{0})$ = $\ \ \ $...($1 $)
$=a(2t)+b(1)+c(0)$
$=2at+b$
