Let $f(x)=\{\begin{array}{cc}{x}^{\frac{3}{2}}sin(\frac{1}{x})\text{;},& (x0)\\ 0\text{;},& (x=0)\end{array}$

$i.$ Calculate $f^{\text{'}}(0)$ using the definition. Hint: Use Example $2$ below $on$ the derivative $of$

$xsin\frac{1}{x}.$

$ii.$ Calculate $f^{\text{'}}(0)$ for $x0.$ Hint: Use the product rule and the chain rule.

iii. $Is{f}^{\text{'}}(x)$ continuous $atx=0?$ Hint: Use Example $2$ below $on$ the derivative $of$

$xsin\frac{1}{x}.$

Example $2$

Let

$f(x)=\{\begin{array}{ccc}xsin(\frac{1}{x})\text{;}& if& x0\\ 0\text{;}& if& x=0\end{array}$

Show that $f(x)is$ continuous.

$Ifx0,f(x)$ has $no$ bad point. $In$ fact, $itis$ the product $of$ two continuous functions $x$

and $sin(\frac{1}{x}),$ both $of$ which are continuous away from $0.$

$At0,$ there $is$ a problem due $to$ the denominator being $0.We$ show that the function, $as$

defined, $is$ still continuous $at0,$ using sequential continuity: xlongrightarrow$0$Longrightarrowf$(x)longrightarrowf(0).$

$In$ fact,

$\underset{\text{x}\text{l}\text{o}\text{n}\text{g}\text{r}\text{i}\text{g}\text{h}\text{t}\text{a}\text{r}\text{r}\text{o}\text{w}0}{lim}f(x)=\underset{\text{x}\text{l}\text{o}\text{n}\text{g}\text{r}\text{i}\text{g}\text{h}\text{t}\text{a}\text{r}\text{r}\text{o}\text{w}0}{lim}xsin(\frac{1}{x})$

$=0sin(\frac{1}{x})xf(0)=0,we$ have

$\underset{\text{x}\text{l}\text{o}\text{n}\text{g}\text{r}\text{i}\text{g}\text{h}\text{t}\text{a}\text{r}\text{r}\text{o}\text{w}0}{lim}f(x)=0=f(0),$

and $so$ the function $is$ continuous $at0.$