$$ Task $2.$

a$.$ Propose a better technique for numerical differentiation that provides higher accuracy than the methods you have worked so far. You need to

write a function for your proposed technique.

Let,

$f(x)=x^5+2\mathrm{.}5x^4-2x^3-6x^2+x+2$

b$.$ Using your proposed method, what is the slope of $f(x)$ at $x=0,-1\mathrm{.}18625$ and step size $=0\mathrm{.}1?$

Compare the error of your method with actual, forward, backward and central differentiation at $x=0,-1\mathrm{.}18625$ by showing in Pandas

Dataframe.

d$.$ Plot actual derivative, Forward derivative, Backward derivative, Central derivative and the derivative from your proposed method in a graph.

Here,

$h=0\mathrm{.}1,-2x1\mathrm{.}2$

$[]$ #$2$a$.$ This cell should not have any output.

Proposed$\_$Method$\_$Name $=""$

#Write Code here

$[]$ #$2$b$.$ This cell should print

$[]$ #$2$c$.$This cell should print

def compare$1($f$,$ f$\_$prime, h$,$ x$)$:

Result $=\{\text{'}$x$\text{'}$ : $[],$ "Actual" : $[],"$FD$"$: $[],"$BD$"$ : $[],"$CD$"$: $[],"$RE$"$: $[],"$FD Error" : $[],"$BD Error": $[],"$CD Error" : $[],"$RE Error" : $[]\}$

#Write code here

compare$1($WUrite code here$)$

$[]$ #$2.$d This cell should plot a graph.