MATLAB: How to differentiate a piecewise function

piecewise function derivative

Hi to all. I have a piecewise function and I want to differentiate it but the derivative will not exist at endpoints. I tried to interpolate it such that the edges are more smooth and the derivative is continuous, but when I plot it, I still get the harsh edges. What to do now? This is my code:
t=0:60;
L =(15-t/2).*(t>=0 & t<=20)+...
   (5).*(t>20 & t<=40 )+(-15+t/2).*(t>40 & t<=60 );
P=interp1(t,L,'pchip');
hold on
plot (t,P,'linewidth',4)
xlim([0 70])
ylim([0 20])

Best Answer

Since you know the formula, you have an advantage - you can just use the known derivative:
t=0:60;
L =(15-t/2).*(t>=0 & t<=20)+...
  (5).*(t>20 & t<=40 )+(-15+t/2).*(t>40 & t<=60 );
P=interp1(t,L,'pchip');
% Plot L vs. t
subplot(3, 1, 1);
plot (t, L, 'LineWidth', 4)
xlim([0 70])
ylim([0 20])
grid on;
xlabel('t', 'FontSize', 20);
ylabel('L', 'FontSize', 20);
% Plot P vs. t
subplot(3, 1, 2);
plot (t, P, 'LineWidth', 4)
xlim([0 70])
ylim([0 20])
grid on;
xlabel('t', 'FontSize', 20);
ylabel('P', 'FontSize', 20);
% Since we know the formula and when it starts and stops each piece
% we can compute the derivative analytically:
dLdt = zeros(1,length(L));
range1 = t>=0 & t<=20;
dLdt(range1) = -0.5;
range2 = t>40 & t<=60;
dLdt(range2) = 0.5;
% Plot dLdt vs. t
subplot(3, 1, 3);
plot (t, dLdt, 'b^-', 'LineWidth', 2)
xlim([0 70])
grid on;
xlabel('t', 'FontSize', 20);
ylabel('dLdt', 'FontSize', 20);
ax = gca;
ax.XAxisLocation = 'origin';
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