Consider the function f$($x$)=$ x $3$ sin$($x$)$ a$.$ Find the exact value of the derivative f $(0\mathrm{.}5).$ b$.$ Estimate f $(0\mathrm{.}5)$ using the central difference formula with h $=0\mathrm{.}1,0\mathrm{.}05,0\mathrm{.}025,$ giving the absolute error for each case. Show that the order of the approximation is O$($h $2).$ c$.$ Estimate f $(0\mathrm{.}5)$ using Richardson extrapolation with h$1=0\mathrm{.}1$ and h$2=0\mathrm{.}05.$ What is the absolute error? d$.$ Estimate f $(0\mathrm{.}5)$ using Richardson extrapolation with h$1=0\mathrm{.}1,$ h$2=0\mathrm{.}05,$ h$3=0\mathrm{.}025.$ What is the absolute error?