1. Derivatives and Differential Equations

2. Differentiation
Differential change
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3. Derivative Definition
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4. Taylor Series
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5. Taylor Series Graphically
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6. Numerical Differentiation Based on Taylor Series
Forward finite-divided difference
Backward finite-divided difference
Centered finite-divided difference
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7. Forward Finite-Divided Difference Approximation
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8. Forward Finite-Difference
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9. Backward Finite-Divided Difference Approximation
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10. Backward Finite-Difference
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11. Centered Finite-Divided Difference Approximation
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12. Centered Finite-Difference
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13. Unequally Spaced Data
One way to calculated derivatives of unequally spaced data is to determine a polynomial fit and take its derivative at a point.
As an example, using a second-order Lagrange polynomial to fit three points and taking its derivative yields:
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14. Derivatives and Integrals for Data with Errors
Numerical differentiation amplifies data errors.
Solution: fit a smooth, differentiable function to the data and take the derivative of the function.
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15. Numerical Differentiation with MATLAB
MATLAB has two built-in functions to help take derivatives, diff and gradient:
diff(x)
Returns the difference between adjacent elements in x. Not the same size as vector x.
diff(y)./diff(x)
Returns the difference between adjacent values in y divided by the corresponding difference in adjacent values of x
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16. Numerical Differentiation, function of a single variable
fx = gradient(f, h)
Gradient can also be used to find partial derivatives for matrices: [fx, fy] = gradient(f, h)
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17. Numerical Differentiation, function of a two variables
To generate the components of a derivative of a function of two variables x, y use
[fx, fy] = gradient(f, h)
Where h scales the magnitude of vectors displayed
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