Derivative of a complex function wrt x_i

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SUMMARY

The discussion focuses on finding the derivative of the function defined as the sum of squared differences between a constant term \( b_{ij} \) and the Euclidean distance \( euclid(x_i, x_j) \) for \( j \) ranging from 0 to \( m \). The specific function is expressed as \( \sum_{j=0}^m(b_{ij}-euclid(x_i, x_j))^2 \). To solve for the derivative with respect to \( x_i \), participants suggest expanding the sum and differentiating each term individually, emphasizing the importance of handling the dimensionality and the sum notation correctly.

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  • Understanding of calculus, specifically differentiation techniques.
  • Familiarity with Euclidean distance calculations in 1D and 2D.
  • Knowledge of summation notation and its implications in mathematical expressions.
  • Basic experience with functions and their derivatives in multivariable calculus.
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  • Study the rules of differentiation for functions involving sums and products.
  • Learn about the properties of Euclidean distance in various dimensions.
  • Explore examples of differentiating complex functions in multivariable calculus.
  • Investigate applications of derivatives in optimization problems.
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Students and professionals in mathematics, particularly those studying calculus and optimization, as well as data scientists and engineers working with distance metrics in machine learning algorithms.

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Given a function
[tex] \sum_{j=0}^m(b_{ij}-euclid(x_i, x_j))^2,[/tex]
where euclid(x_i, x_j) denotes the Euclidean distance (1D or 2D) between x_i and x_j.
I'm supposed to find the derivative with respect to x_i.
The sum sign and the dimensionality are the problem for me.
Any help on how to solve this is appreciated.
 
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try writing out the sum to help

[tex]\sum_{j=0}^m(b_{ij}-euclid(x_i, x_j))^2 = (b_{i1}-euclid(x_i, x_1))^2 + (b_{i2}-euclid(x_i, x_2))^2 +..+(b_{ii}-euclid(x_i, x_i))^2+..+(b_{im}-euclid(x_i, x_m))^2[/tex]

now differentiate term by term
 

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