$\{$USE PYTHON ONLY $\}$

Let $f(x)$ be a function of $x.$

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

a$.$ Plot the function for

$-2\mathrm{.}5x1\mathrm{.}5$

b$.$ What is the actual slope of $f(x)$ at $x=0,-1\mathrm{.}18625?$ Print $f^{\text{'}}(x)$ and plot $f^{\text{'}}(x)$ at $-2x1\mathrm{.}2.$

For c to e$,$ assume step size is $0\mathrm{.}1.$

c$.$ Use forward differntiation to figure out the slope at $x=0,-1\mathrm{.}18625.$

d$.$ Use backward differntiation to figure out the slope at $x=0,-1\mathrm{.}18625.$

e$.$ Use central differntiation to figure out the slope at $x=0,-1\mathrm{.}18625.$

f$.$ Compare the error of each method with actual differentiation at $x=0,-1\mathrm{.}18625$ by showing in a Pandas Dataframe.

g$.$ plot error vs h curves with proper label and color for each method at $x=0$ and $h=[0\mathrm{.}55,0\mathrm{.}3,\mathrm{.}17,0\mathrm{.}1,0\mathrm{.}055,0\mathrm{.}03,0\mathrm{.}017,0\mathrm{.}01].$