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for the first derivative, for the second derivative, for the third derivative, and for the nth derivative. When f is a function of several variables, it is common to use "∂", a stylized cursive lower-case d, rather than "D". As above, the subscripts denote the derivatives that are being taken.
Image derivatives can be computed by using small convolution filters of size 2 × 2 or 3 × 3, such as the Laplacian, Sobel, Roberts and Prewitt operators. [1] However, a larger mask will generally give a better approximation of the derivative and examples of such filters are Gaussian derivatives [2] and Gabor filters. [3]
The higher-order derivatives are less common than the first three; [1] [2] thus their names are not as standardized, though the concept of a minimum snap trajectory has been used in robotics and is implemented in MATLAB. [3] The fourth derivative is referred to as snap, leading the fifth and sixth derivatives to be "sometimes somewhat ...
Differential quadrature is the approximation of derivatives by using weighted sums of function values. [22] [23] Differential quadrature is of practical interest because its allows one to compute derivatives from noisy data.
First-order means that only the first derivative of y appears in the equation, and higher derivatives are absent. Without loss of generality to higher-order systems, we restrict ourselves to first-order differential equations, because a higher-order ODE can be converted into a larger system of first-order equations by introducing extra variables.
Codes which calculate exactly electromagnetic scattering by an aggregate of spheres in a single orientation or at an average over individual orientations. 2013 MSTM D. W. Mackowski [19] Fortran Codes which calculate exactly electromagnetic scattering by an aggregate of spheres and spheres within spheres for complex materials.
To get the coefficients of the backward approximations from those of the forward ones, give all odd derivatives listed in the table in the previous section the opposite sign, whereas for even derivatives the signs stay the same. The following table illustrates this: [5]
The first derivative of a function f of a real variable at a point x can be approximated using a five-point stencil as: [1] f ′ ( x ) ≈ − f ( x + 2 h ) + 8 f ( x + h ) − 8 f ( x − h ) + f ( x − 2 h ) 12 h {\displaystyle f'(x)\approx {\frac {-f(x+2h)+8f(x+h)-8f(x-h)+f(x-2h)}{12h}}}
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