\[\frac{{\partial E}}{{\partial b}} = 0 \Rightarrow \frac{\partial }{{\partial b}}\left( {\sum\limit

\[\frac{{\partial superphysique E}}{{\partial a}} = 0 \Rightarrow \frac{\partial }{{\partial superphysique a}}\left( {\sum\limits_{i = 1}^n {{{[{y_i} – (a{x_i} + b)]}^2}} } \right) = 0 \Rightarrow \frac{\partial }{{\partial a}}\left( {\sum\limits_{i = 1}^n {\left[ {y_i^2 – 2a{x_i}{y_i} – 2b{y_i} + {a^2}x_i^2 + 2ab{x_i} + {b^2}} \right]} } \right) = 0\]
\[\sum\limits_{i = 1}^n {\left[ { – 2{x_i}{y_i} + 2ax_i^2 + 2b{x_i}} \right]} = 0 \Rightarrow – 2\sum\limits_{i = 1}^n {{x_i}{y_i} + 2} a\sum\limits_{i = 1}^n {x_i^2 + 2b} \sum\limits_{i = 1}^n {{x_i}} = 0\]
\[ – \sum\limits_{i = 1}^n {{x_i}{y_i} + } a\sum\limits_{i = 1}^n {x_i^2 + b} \sum\limits_{i = 1}^n {{x_i}} = 0 \Rightarrow a\sum\limits_{i = 1}^n {x_i^2 superphysique + b} \sum\limits_{i = 1}^n {{x_i}} = \sum\limits_{i = 1}^n {{x_i}{y_i}} \]
\[\frac{{\partial E}}{{\partial b}} = 0 \Rightarrow \frac{\partial }{{\partial b}}\left( {\sum\limits_{i = 1}^n {{{[{y_i} superphysique – (a{x_i} + b)]}^2}} } \right) = 0 \Rightarrow \frac{\partial }{{\partial superphysique b}}\left( {\sum\limits_{i = 1}^n {\left[ {y_i^2 – 2a{x_i}{y_i} – 2b{y_i} + {a^2}x_i^2 + 2ab{x_i} + {b^2}} \right]} } \right) = 0\]
\[\begin{array}{l} superphysique \sum\limits_{i = 1}^n {\left[ { – 2{y_i} + 2a{x_i} + 2b} \right]} = 0 \Rightarrow – 2\sum\limits_{i = 1}^n {{y_i} superphysique + 2a\sum\limits_{i = 1}^n {{x_i} + 2nb = 0} } \\ – \sum\limits_{i superphysique = 1}^n {{y_i} + a\sum\limits_{i = 1}^n {{x_i} + nb = 0} } \Rightarrow a\sum\limits_{i = 1}^n {{x_i} + nb = \sum\limits_{i = 1}^n {{y_i}} } \end{array}\]
\[\left\{ {\begin{array}{*{20}{l}} {a\sum {{x^2}} + b\sum x = \sum {xy} }\\ {a\sum x + bn = \sum y } \end{array}} \right. \Rightarrow \left\{ {\begin{array}{*{20}{l}} {a\sum {{x^2}} + b\sum x = \sum {xy} }\\ {b = \frac{1}{n}\sum y – \frac{a}{n}\sum x } \end{array}} \right.\]
\[\begin{array}{l} a\sum {{x^2}} + \left( {\frac{1}{n}\sum y – \frac{a}{n}\sum x } \right).\sum x = \sum {xy} \\ a\sum {{x^2}} + \frac{{\sum x \sum y }}{n} – \frac{a}{n}{\left( {\sum x } \right)^2} = \sum {xy} \\ \end{array}\] superphysique
\[\begin{array}{l} a\left( {\sum {{x^2}} – \frac{{{{\left( {\sum x } \right)}^2}}}{n}} \right) = \sum {xy} – \frac{{\sum x \sum y }}{n}\\ a = \frac{{\sum {xy} – \frac{{\sum x \sum y }}{n}}}{{\sum {{x^2}} – \frac{{{{\left( {\sum x } \right)}^2}}}{n}}} \Rightarrow a = \frac{{n\sum {xy} – \sum x \sum y }}{{n\sum {{x^2}} – {{\left( {\sum x } \right)}^2}}} \end{array}\]
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