Take you same derivate function from last week and let's make it more general by passing in f$($x$),$ and x

You derivative function will now look something like this my$\_$der$\_$calc$($ f$,$x$,$N$,$eps, option$)$

eps is a tolerance for determining the accuracy of your derivatives. We will use this later

To test, let's use our function from last laboratory class, x$(1-$x$)$

The output argument, df$,$ should be the numerical derivatives computed for x according to the method

defined by the input argument, option

Remember, the forward difference method "loses" the last point, the backward difference method loses the

first point, and the central difference method loses the first and last points. So you have use some

interpolation to get these points. Try linear

By obtaining the derivative at the first and last point, you will have the exact same number of values for

both df and x$.$ Although, you could set them to same value as well, however, your plot will only be linear

for the interior points

Plot df$.$vs x$,$ notice the functional form of your plot.

Compare with the actual form with the derivative of the given function, if the calculated derivative is not

within $1\backslash$times $10^(-3)$ of the given function derivative add more points until the calculated derivative is less

than or equal to $1\backslash$times $10^(-3)$ of the given function's derivative

Next, add to you derivative function the capability to check for accuracy by comparing maximum errors between successive calculations of the derivative. You will need a while loop that checks when the maximum error is $<=$ eps. To find the maximum array create a numpy array in your derivative function. Call it "maxe" where each element of the array would contain $|$d$$j$+($x;$)$ d$^3($xi$)|.$ j $+1$ refers to the finer grid where the j $+1$ grid would use twice as many points as the j grid. For example, j $=1($the starting grid$)$ would have N $=10$ points, and the j $+1$ grid would use $2$N $=20$ points, the j $+2$ grid would use $40$ points and so on$.$ Now for the $1$st pass through the while$-$loop you cannot converge you haven't started the comparison. So construct you code to avoid this situation. For this part of the laboratory, let's use for our function, f$($x$)=$ x logx