Find-the-partial-derivatives-for-each-of-the-following-a-Z-3x-2-5x-7y-2-b-Z-w-x-y-2-3w-2x-4y-

Question Number 18470 by tawa tawa last updated on 22/Jul/17

Find the partial derivatives for each of the following

(a) Z = {3 x}^{2} {(5 x + 7 y)}^{2}

(b) Z = {(w - x - y)}^{2} (3 w + 2 x - 4 y)

Answered by ajfour last updated on 22/Jul/17

(a)

\frac{\partial z}{\partial x} = 6 x {(5 x + 7 y)}^{2} + {3 x}^{2} [2 (5 x + 7 y) (5)]

= 6 x (5 x + 7 y) (10 x + 7 y)

\frac{\partial z}{\partial y} = {3 x}^{2} (2) (5 x + 7 y) (7)

= {42 x}^{2} (5 x + 7 y)

(b) \frac{\partial z}{\partial x} = - 2 (w - x - y) (3 w + 2 x - 4 y)

+ 2 {(w - x - y)}^{2}

= - 2 (w - x - y) (2 w + 3 x - 3 y)

\frac{\partial z}{\partial y} = - 2 (w - x - y) (3 w + 2 x - 4 y)

- 4 {(w - x - y)}^{2}

= - 2 (w - x - y) (5 w - 6 y)

\frac{\partial z}{\partial w} = 2 (w - x - y) (3 w + 2 x - 4 y)

+ 3 {(w - x - y)}^{2}

= (w - x - y) (9 w + x - 11 y) .

Commented by tawa tawa last updated on 22/Jul/17

God bless you sir .