Differentiation question using diff and polyder

I already wrote my code, I just need help figuring out a couple of errors.

So my function has 3 parts. For the first part, I want to crete a function that finds the derivative of ln x. I want to set x = 2:0.05:4. In the function MyDierentiation, I want to compute the approximated values of derivatives at x by using the MATLAB function diff, and assign the value to df1

function [df1, df2] = MyDifferentiation()

%%Part A

x = 2:0.05:4
y= log(x)
df1= diff(x)

I am getting an error in my code. My instructor told me that if f'(x) is a 1x21 array, df1 will be a 1x20 array. So, i'm guessing I have to make df1 have the same length as x by assuming that the last point has the same derivative as the preceding point? Is that right? how would I go about doing this on MATLAB?

%%Part B

x = 2:0.05:4
y=log(x)
df2= polyfit(x,y)
df2=polyder(x)

I also compute the approximated derivative using the function polyder (By using the same x = 2:0.05:4 and computing the corresponding y values of the function y = ln(x).) I also wan to fit the x and y data by a 3rd order polynomial. The values of the derivative at x will be assigned to df2.

%%Part C

hold on
plot (df1) 
plot (df2)
hold off
ylabel('derivative')
xlabel('x')
legend('diff','polyder','exact values')
lines = {'diff','polyder','exact values'}
legend(lines)

I haven't tested this code because of the errors above, but hopefully it's plot the right graph once the errors are fixed. ( I wanted to plot df1, df2, and the exact values of the derivative in a single plot. Please note that I am using a blue solid line for df1, red dashed line for df2, and green dotted line for exact values. I am also setting LineWidth to 2 and providing a legend to the graph. )

thank you everyone!

Answer 1

[Star Strider]

Problems like this are in a sense a bit artful because you have options. I suggest:

% % Part A:

Don't you actually want to calculate:

df1 = diff(y);

instead?

In that situation, I suggest perhaps plotting df1 as:

plot(x(1:length(df1)), df1)

% % Part B:

If you want to calculate a 3rd order polynomial fit, I suggest:

df2 = polyfit(x,y,3);

You might also find the polyval function helpful.

[Ben]

oh, i see. Thank you! I tried testing the function by calling

[df1, df2] = MyDifferentiation(); 
df1(1)

but i keep getting .0247 instead of 0.4939. is there a reason for this?

[Star Strider]

My pleasure!

The reason is that the derivative is defined as df/dx, so this:

dfdx1 = diff(log(x))./diff(x);

should come close to the answer you are looking for.

If you want a more precise numerical answer to this (and other) derivatives, you can have fun with this idea:

derfx = @(f,x,h) (f(x+h)-f(x))/h;
x = 2:0.05:4;
derlog = derfx(@log, x, 1E-10);

That code snippet uses Anonymous Functions.

[Ben]

this is so cool! matlab is really like a super-calculator. Thanks!!

[Star Strider]

My pleasure! I'm glad it worked for you.