ex a PartialDerivatvessarti at derivative symbol diff The partial derivative of written treated as alone f f x y with respect to x a constant and fixy is considered a function the partial derivative with respectto g Similar the derivative of flxy where y is for is x Let fix g 5 392 Compute b If fix y Ey 3 5 y That 5 5 29 10 variable Any 5 32 Sy x cons 7 variable 594 3 2 1546 ex Let fly g I ix g muariables 3 2 2 5 29 0 6 624 stants 3 72 9 5g Compute b ariable If 5 2X 0 256 ex compute It and If for each b fixg e c fix g y y fix g 4 39 518 8 4 39 51714 32 4 39571 814 34 517 3 2414 34550 y Ey 2xye 1 3 4411 i 1 344 41 1 Fy ex Let f x y a Calculate If 6 29 1 4 611 214 6 8 3 2 2xy 59 1 4 b Evaluate at x y 1 4 Fy 2 5 1 4 5 2 1 2 5 to Vow that we understand the algebraic way find partial derivatives what does this look like siiiiiii iiiiiii is the ordinary derivative with y held constant Since ate of change of flx y with respect to x I interpret the partial derivatives from ex 4 above for y held constant It gives th ixg 3 42 9 59 1 4 14 varies if y is held constant at 4 nd hanges at a rate 14 times hange of X then fix y the if x increases by h units 14h units e flag increases by 7 varies then fix y 511 4 if y is held constant at 4 and changes at a rate 7 times change of X ie if x increases by h units 7h units flag increases by the ft I small we have function of 2 variables Then if h and K are f f ath b a btk f a b f a b If a b a b K t ex Let fly y z compute x yZ 32 If and 2 2 42 Rz Ry 3 Fy 2 b calculate Fz 2 3 1 Ry 3112 3 1 22 3 46 3 12 3 3 ex Let x y z It Ily 152 10 7 5 fly g 2 be the heat loss function from 7 1 Hxy 15 2 Calculate interpret 14yz 10 7 5 1515 75 11 7 77 1520 loss with marginal heat respect to change in x x is changed from 10 by h units if and the values of yo 2 remain fixed the amount of heat loss at 7 and 5 will change by 152 h units ex consider the production function fix g 60 y which units gives the number of units of goods produced when t a find labor and y units of capital are used 601 45 Evaluate II 81 16 and If x y y It and If 45 81514 16 YETI 4552 If 6014 y 314 3 4 15 at 81 y 16 81 167 15 8154 165 15 1 40 50 c Interpret the numbers computed in part b marginal productivity of labor and marginal productivity of capital Just higher order variables like computing second derivatives and derivatives of two we can do the same with variable in one differenttypes partial derivative of It partial derivative of partial derivative of partial derivative of It in respect to in respect to y in respect to in respect to y IF f If Almost all functions hold true If ex t Let fix g ex I Find 2x yex 2 b d y Ey 2xye 9 12 9 Re Y 2 e 9 iy 2x x e x 2 ye9 e ezxefylfy tl 2x