Best solutions to - "Use Equations 7 to find ∂z∂x and ∂z∂y.ez=xyz" - Plainmath

boske9s

Open question

2022-08-18

Use Equations 7 to find

\frac{\partial z}{\partial x} and \frac{\partial z}{\partial y}

.

e^{z} = x y z

Answer & Explanation

carponih7

Beginner2022-08-19Added 13 answers

e^{z} = x y z

e^{z} - x y z = 0

F (x, y, z) = e^{z} - x y z = 0

According to formula 7
Suppose that an equation F(x,y,z)=0 defines z Implicitly in terms of x and y
Then

\frac{\partial z}{\partial x} = - \frac{F_{x}}{F_{z}} \frac{\partial z}{d y} = - \frac{F_{y}}{F_{z}}

F_{x} = - y z

F_{y} = - x z

F_{z} = e^{z} - x y

Therefore

\frac{\partial z}{d x} = - \frac{F_{x}}{F_{z}} = - \frac{- y z}{e^{z} - x y} = \frac{y z}{e^{z} - x y}

\frac{\partial z}{d y} = - \frac{F_{y}}{F_{z}} = - \frac{- x z}{e^{z} - x y} = \frac{x z}{e^{z} - x y}

Result:

\frac{\partial z}{d x} = \frac{y z}{e^{z} - x y} \frac{\partial z}{d y} = \frac{x z}{e^{z} - x y}

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