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

$$\mathrm{Find}\mathrm{the}\mathrm{partial}\mathrm{derivatives}\mathrm{for}\mathrm{each}\mathrm{of}\mathrm{the}\mathrm{following}$$$$\left(\mathrm{a}\right)\mathrm{Z}={3\mathrm{x}}^2{(5\mathrm{x}+7\mathrm{y})}^2$$$$\left(\mathrm{b}\right)\mathrm{Z}={(\mathrm{w}-\mathrm{x}-\mathrm{y})}^2(3\mathrm{w}+2\mathrm{x}-4\mathrm{y})$$

Answered by ajfour last updated on 22/Jul/17

$$\left(\mathrm{a}\right)$$$$\frac{\partial \mathrm{z}}{\partial \mathrm{x}}=6\mathrm{x}{(5\mathrm{x}+7\mathrm{y})}^2+{3\mathrm{x}}^2\left[2(5\mathrm{x}+7\mathrm{y})\left(5\right)\right]$$$$=6\mathrm{x}(5\mathrm{x}+7\mathrm{y})(10\mathrm{x}+7\mathrm{y})$$$$\frac{\partial \mathrm{z}}{\partial \mathrm{y}}={3\mathrm{x}}^2\left(2\right)(5\mathrm{x}+7\mathrm{y})\left(7\right)$$$$={42\mathrm{x}}^2(5\mathrm{x}+7\mathrm{y})$$$$$$$$\left(\mathrm{b}\right)\frac{\partial \mathrm{z}}{\partial \mathrm{x}}=-2(\mathrm{w}-\mathrm{x}-\mathrm{y})(3\mathrm{w}+2\mathrm{x}-4\mathrm{y})$$$$+2{(\mathrm{w}-\mathrm{x}-\mathrm{y})}^2$$$$=-2(\mathrm{w}-\mathrm{x}-\mathrm{y})(2\mathrm{w}+3\mathrm{x}-3\mathrm{y})$$$$\frac{\partial \mathrm{z}}{\partial \mathrm{y}}=-2(\mathrm{w}-\mathrm{x}-\mathrm{y})(3\mathrm{w}+2\mathrm{x}-4\mathrm{y})$$$$-4{(\mathrm{w}-\mathrm{x}-\mathrm{y})}^2$$$$=-2(\mathrm{w}-\mathrm{x}-\mathrm{y})(5\mathrm{w}-6\mathrm{y})$$$$\frac{\partial \mathrm{z}}{\partial \mathrm{w}}=2(\mathrm{w}-\mathrm{x}-\mathrm{y})(3\mathrm{w}+2\mathrm{x}-4\mathrm{y})$$$$+3{(\mathrm{w}-\mathrm{x}-\mathrm{y})}^2$$$$=(\mathrm{w}-\mathrm{x}-\mathrm{y})(9\mathrm{w}+\mathrm{x}-11\mathrm{y}).$$

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

$$\mathrm{God}\mathrm{bless}\mathrm{you}\mathrm{sir}.$$

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