Could someone please check my work using partial differential equations

Let (x1, y1), (X2, y2), . .. , (m, ym) be given points by experimental ob- Find the formula for a and b. servation. Let y = ax2 + b be the best fit curve by least square criterion. Assume f(x) = axa+b is the best fit for the given date by Least - squares criterion. Find the formula for a and b. minimize | 4 . - f ( x , ) 1 2 + 1 Y2 - f ( x 2 > / 2 + ... ( ym - f(xm) | 2 i. e . = Minimize E ( 4 ; - f ( x ; ) ) =S a z x z x + b ( z x 2 ) * = Ex - Y zx 2 4 LX SUM a E x " E x + mb E x = Ey Ex " > minimize ( =( yi taxi + by ) ' - ( y ; - ax; - by " b ( z x - ) 2 - MbEx " = Ex # y zx 2 - Eyex4 -7 6 [ ( E x 7 ) 2 - MEX " J = Ex- y Ex " - Eyex " To Minimize S, we set these partial derivatives = D: Ex YEX ? - Zy EX derivative by chain rule by distribution 7 b = ( EX 2 ) 2 - MEX 4 Co = 2 2 ( 4; - ax; - b) . (- xi ) = D = 22 - xiyitaxi + by; =0=> Exi yi Ex - EY ; EX . " dividing both sides by 2 M -> b = A M FEWN E - x; y; + a x; + b x; = > > - E x; y; + a z x; + bz x; = Q=> ( EXi ) ' - MEX; M 2 2 x; +bz x. = z X; Yi A derivative by chan rule distribure ' Substituting 6 into (3) equation: as = = 2 (4 ; - ax; - b) . (-1) = 0 => 22- Yitax; +b=1 ab a z x t m z x y Ex - Ey Ex - = EY dividing both sides by ? ( Ex ) * - mex* M 2 2 - 4; + ax; + b = Q =7 - zy; + a Ex; + mb = D => 2 = 1 Multiplying both sides by EX 2 . . azxi +mb = I WI - yi Ex y Ex - EYEX" = zy a + x 3 ( Ex 2) 2 - m Ex 4 EX a z x + b Ex. = E xy @ multiply is equation by zx- 2x y Ex " - Ey Ex ( Ex 2) 2 - m Ex 4 a +mb = Ey 2 multiply 2 equation by Ex." EWA E Xi Y ; EX ; - Z EW ( Exi ) 2 - ME