Question

use the equations
\frac { partial z } { partial x } = - \frac { \frac { partial f } { partial x } } { \frac { partial f } { partial z } } \text { and } \frac { partial z } { partial y } = - \frac { \frac { partial f } { partial y } } { \frac { partial f } { partial z } }
to find ( \frac { partial z } { partial x } ) and ( \frac { partial z } { partial y } ).
x ^ { 2 } - y ^ { 2 } + z ^ { 2 } - 6 z = 7
\frac { partial z } { partial x } =
\frac { partial z } { partial y } =

Explanation:

Step1: Define \( F(x,y,z) \)

Let \( F(x,y,z)=x^{2}-y^{2}+z^{2}-6z - 7\).

Step2: Calculate partial derivatives

• Calculate \(\frac{\partial F}{\partial x}\):

Using the power rule \(\frac{\partial}{\partial x}(x^{n})=nx^{n - 1}\), \(\frac{\partial F}{\partial x}=2x\).

• Calculate \(\frac{\partial F}{\partial y}\):

Using the power rule, \(\frac{\partial F}{\partial y}=-2y\).

• Calculate \(\frac{\partial F}{\partial z}\):

Using the power rule, \(\frac{\partial F}{\partial z}=2z-6\).

Step3: Find \(\frac{\partial z}{\partial x}\)

Substitute into the formula \(\frac{\partial z}{\partial x}=-\frac{\frac{\partial F}{\partial x}}{\frac{\partial F}{\partial z}}\).
\(\frac{\partial z}{\partial x}=-\frac{2x}{2z - 6}=\frac{x}{3 - z}\).

Step4: Find \(\frac{\partial z}{\partial y}\)

Substitute into the formula \(\frac{\partial z}{\partial y}=-\frac{\frac{\partial F}{\partial y}}{\frac{\partial F}{\partial z}}\).
\(\frac{\partial z}{\partial y}=-\frac{-2y}{2z - 6}=\frac{y}{3 - z}\).

Answer:

\(\frac{\partial z}{\partial x}=\frac{x}{3 - z}\), \(\frac{\partial z}{\partial y}=\frac{y}{3 - z}\)