consider the function f(x) = cos(x)/(x^3) + e^x. numerically compute its derivative at x = 4 using the first order finite difference (backward) method in matlab

To compute the derivative of a function numerically at a specific point using the first-order backward finite difference method in MATLAB, you can follow these steps:

1. Define the function f(x) in MATLAB. In this case, the function is f(x) = cos(x)/(x^3) + exp(x).

   main.m
   f = @(x) cos(x)./(x.^3) + exp(x);
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2. Define the point at which you want to calculate the derivative. In this case, x = 4.

   main.m
   x = 4;
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3. Define the step size, h, used in the finite difference method. This determines the spacing between the points used to calculate the derivative. For the backward method, h should be a small negative value.

   main.m
   h = -0.001;
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4. Calculate the derivative at x using the first-order backward finite difference formula:

   main.m
   df_dx = (f(x) - f(x+h))/h;
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5. Display the result.

   main.m
   disp(['The derivative of f(x) at x = 4 is approximately: ' num2str(df_dx)]);
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Putting it all together, the MATLAB code to numerically compute the derivative of f(x) = cos(x)/(x^3) + exp(x) at x = 4 using the first-order backward finite difference method would be:

main.m
f = @(x) cos(x)./(x.^3) + exp(x);
x = 4;
h = -0.001;

df_dx = (f(x) - f(x+h))/h;

disp(['The derivative of f(x) at x = 4 is approximately: ' num2str(df_dx)]);
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Please note that the choice of h affects the accuracy of the numerical derivative. A smaller h will generally yield a more accurate result, but it also introduces numerical errors and round-off errors. It is important to choose an appropriate value for h based on the problem and desired level of accuracy.