Use given derivative.m MATLAB code $($listed below$)$ to add both the centered difference and Richardson extrapolation methods of numerical differentiation. They need to be subfunctions like forwarddiff. Backward difference is already implemented $($forwarddiff with a negative h$).$ Use structures to implement the methods, so refer to forwarddiff for an example of what the formulas for centered difference and Richardson extrapolation should look like.

derivative.m:

function approx $=$ derivative$($f$,$ x$,$ varargin$)$

$\%$DERIVATIVE computes numerical derivative using chosen method:

$\%$ fwd diff, bkwd diff, centered diff,

$\%$ and Richardson extrapolation.

$\%$ Syntax: approx $=$ derivative$($f$,$ x$,$ 'Method', 'forward', 'StepSize', $1$e$-4)$

$\%$ Inputs:

$\%$ f function handle

$\%$ x array of values to compute the numerical derivative

$\%---$Optional Arguments$---$

$\%$ field name name of target parameter 'Method', 'StepSize'

$\%$ field parameter value 'forward', 'backward', 'centered',

$\%$ 'Richardson', for field name 'Method'. Default method

$\%$ is forward difference.

$\%$

$\%$ floating point step size h for field name 'StepSize'. The

$\%$ default step size depends on the difference method.

$\%$ Outputs:

$\%$ approx the numerical approx. to the derivative of function

$\%$ Example:

$\%$ zero $=$ derivative$($@cos$,$ pi$/3)$

$\%$ zero $=$ derivative$($@sin$,$ pi$/2,$ 'Method', 'richardson'$)$

$\%$ zero $=$ derivative$($@$($x$)$ exp$($x$)-1,-1,$ 'stepsize', $1$e$-6)$

$\%$ data structure

data.f $=$ f;

data.x $=$ x;

data.h $=$ sqrt$($eps$)$;

method $=$ @forwarddiff;

$\%$ unknown order of arguments

stepSizeSet $=$ false;

$\%$ varargin has some optional arguments!

if nargin $>2$

$\%$ loop through the field names $($odd indexes$)$

for k$=1$:$2$:length$($varargin$)$

$\%$ compare the field name to constants

switch$($lower$($varargin$\{$k$\}))$

case 'method'

$\%$ use the even $($following$)$ element of varargin for the

$\%$ name of the method

switch$($lower$($varargin$\{$k$+1\}))$

case $\{\text{'}$backward$\text{'},\text{'}$b$\text{'}\}$

method $=$ @forwarddiff;

if ~stepSizeSet

data.h $=-$sqrt$($eps$)$;

end

if data.h $>0$

data.h $=-$data.h;

end

case $\{\text{'}$centered$\text{'},$ 'center', 'central', $\text{'}$c$\text{'}\}$

method $=$ @centered;

if ~stepSizeSet

data.h $=$ eps$^(1/3)$;

end

case $\{\text{'}$richardson$\text{'},\text{'}$r$\text{'}\}$

method $=$ @richardson;

if ~stepSizeSet

data.h $=$ eps$^(1/5)$;

end

end

case char$(\text{'}$stepsize$\text{'})$

$\%$ use the next value in varargin for the floating point

$\%$ step size h

data.h $=$ varargin$\{$k$+1\}$;

stepSizeSet $=$ true;

end

end

end

approx $=$ method$($data$)$;

end

function approx $=$ forwarddiff$($data$)$

$\%$derivate computes a numerical derivative using fwd diff, or bkwd if data.h

$\%<0.$

$\%$ Syntax: approx $=$ fowarddiff$($data$)$

$\%$ Inputs:

$\%$ data structure with following fields:

$\%$ f function handle

$\%$ x array of values to compute the numerical derivative

$\%$ h step size for difference methods

$\%$ Outputs:

$\%$ approx the numerical approximation to the derivative of function

$\%$ Example:

$\%$ zero $=$ forwarddiff$($data$)$

$\%$ f$\text{'}($x$)=($f$($x$+$h$)-$f$($x$))/$h

$\%$ if h$<0,$ then the sign of the denominator will reverse the difference

$\%$ in the numerator.

approx $=($data$.$f$($data$.$x $+$ data.h$)-$ data.f$($data$.$x$))./($data$.$h$)$;

end

function approx $=$ centered$($data$)$

$\%$CENTERED computes numerical derivative using centered diff

$\%$ Syntax: approx $=$ centered$($data$)$

$\%$ Inputs:

$\%$ data structure with the following fields:

$\%$ f function handle

$\%$ x array of values to compute the numerical derivative

$\%$ h step size for difference methods

$\%$ Outputs:

$\%$ approx the numerical approximation to the derivative of function

$\%$ Example:

$\%$ zero $=$ centered$($data$)$

approx $=0$; $\%$ replace this

end

function approx $=$ richardson$($data$)$

$\%$RICHARDSONcomputes numerical derivative using Richardson Extrapolation

$\%$ Syntax: approx $=$ richardson$($data$)$

$\%$ Inputs:

$\%$ data structure with the following fields:

$\%$ f function handle

$\%$ x array of values to compute the numerical derivative

$\%$ h step size for difference methods

$\%$ Outputs:

$\%$ approx the numerical approximation to the derivative of function

$\%$ Example:

$\%$ zero $=$ richardson$($data$)$

approx $=0$; $\%$ replace this

end