Solve x^2 -y^2 -z^2 =1

Question

Solve the equation

Solve for x

Solve for y

Solve for z

x = 1 + y^{2} + z^{2} x = - 1 + y^{2} + z^{2}

Evaluate

x^{2} - y^{2} - z^{2} = 1

Move the expression to the right-hand side and change its sign

x^{2} = 1 + y^{2} + z^{2}

Take the root of both sides of the equation and remember to use both positive and negative roots

x = \pm 1 + y^{2} + z^{2}

Solution

x = 1 + y^{2} + z^{2} x = - 1 + y^{2} + z^{2}

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Find \frac{\partial z}{\partial x} by differentiating the equation directly

Find \frac{\partial z}{\partial y} by differentiating the equation directly

\frac{\partial z}{\partial x} = \frac{x}{z}

Evaluate

x^{2} - y^{2} - z^{2} = 1

Find \frac{\partial z}{\partial x} by taking the derivative of both sides with respect to x

\frac{\partial}{\partial x} (x^{2} - y^{2} - z^{2}) = \frac{\partial}{\partial x} (1)

Use differentiation rule \frac{\partial}{\partial x} (f (x) \pm g (x)) = \frac{\partial}{\partial x} (f (x)) \pm \frac{\partial}{\partial x} (g (x))

\frac{\partial}{\partial x} (x^{2}) - \frac{\partial}{\partial x} (y^{2}) - \frac{\partial}{\partial x} (z^{2}) = \frac{\partial}{\partial x} (1)

Use \frac{\partial}{\partial x} x^{n} = n x^{n - 1} to find derivative

2 x - \frac{\partial}{\partial x} (y^{2}) - \frac{\partial}{\partial x} (z^{2}) = \frac{\partial}{\partial x} (1)

Use \frac{\partial}{\partial x} (c) = 0 to find derivative

2 x - 0 - \frac{\partial}{\partial x} (z^{2}) = \frac{\partial}{\partial x} (1)

Evaluate

More Steps

Evaluate

\frac{\partial}{\partial x} (z^{2})

Use the chain rule \frac{\partial}{\partial x} (f (g)) = \frac{\partial}{\partial g} (f (g)) \times \frac{\partial}{\partial x} (g) where the g = z, to find the derivative

\frac{\partial}{\partial z} (z^{2}) \frac{\partial z}{\partial x}

Find the derivative

2 z \frac{\partial z}{\partial x}

2 x - 0 - 2 z \frac{\partial z}{\partial x} = \frac{\partial}{\partial x} (1)

Removing 0 doesn't change the value,so remove it from the expression

2 x - 2 z \frac{\partial z}{\partial x} = \frac{\partial}{\partial x} (1)

Find the partial derivative

2 x - 2 z \frac{\partial z}{\partial x} = 0

Move the expression to the right-hand side and change its sign

- 2 z \frac{\partial z}{\partial x} = 0 - 2 x

Removing 0 doesn't change the value,so remove it from the expression

- 2 z \frac{\partial z}{\partial x} = - 2 x

Divide both sides

\frac{- 2 z \frac{\partial z}{\partial x}}{- 2 z} = \frac{- 2 x}{- 2 z}

Divide the numbers

\frac{\partial z}{\partial x} = \frac{- 2 x}{- 2 z}

Solution

\frac{\partial z}{\partial x} = \frac{x}{z}

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