$Q6.$ When $we$ differentiate a monomial function like $c{x}^{n}k$ times $we$ might notice a pattern,

$\frac{d}{(d)x}[c{x}^n]=cn{x}^{n-1}$

$\frac{d^2}{(d)x^2}[c{x}^n]=cn(n-1)x^{n-2}$

vdots

$\frac{d^k}{(d)x^k}[c{x}^n]=c\frac{n!}{(n-k)!}{x}^{n-k}=c\frac{(n+1)}{(n-k+1)}{x}^{n-k}$

$By$ analogy, $we$ can use the gamma function $to$ define a "half derivative",

$\frac{d^{\frac{1}{2}}}{(d)x^{\frac{1}{2}}}[c{x}^n]=c\frac{(n+1)}{(n+\frac{1}{2})}{x}^{n-\frac{1}{2}}$

$(a)$ Show that $\frac{d^{\frac{1}{2}}}{(d)x^{\frac{1}{2}}}[x]=2\sqrt[\phantom{2}]{\frac{x}{}}(You$ may need $togo$ rewatch our gamma function lesson$)$

$(b)It$ turns out that applying our "half derivative" twice $is$ the same $as$ just applying the usual

derivative. Verify this $by$ showing that $\frac{d^{\frac{1}{2}}}{(d)x^{\frac{1}{2}}}[\frac{d^{\frac{1}{2}}}{(d)x^{\frac{1}{2}}}[c{x}^n]]=\frac{d}{dx}[c{x}^n].$