Solve x^2 y^2 z^2 = 4

Evaluate

x^{2} y^{2} z^{2} = 4

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} (4)

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

\frac{\partial}{\partial x} (x^{2} y^{2}) z^{2} + x^{2} y^{2} \times \frac{\partial}{\partial x} (z^{2}) = \frac{\partial}{\partial x} (4)

Evaluate

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2 x y^{2} z^{2} + x^{2} y^{2} \times \frac{\partial}{\partial x} (z^{2}) = \frac{\partial}{\partial x} (4)

Evaluate

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2 x y^{2} z^{2} + x^{2} y^{2} \times 2 z \frac{\partial z}{\partial x} = \frac{\partial}{\partial x} (4)

Evaluate

2 x y^{2} z^{2} + 2 z x^{2} y^{2} \frac{\partial z}{\partial x} = \frac{\partial}{\partial x} (4)

Find the partial derivative

2 x y^{2} z^{2} + 2 z x^{2} y^{2} \frac{\partial z}{\partial x} = 0

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

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

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

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

Divide both sides

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

Divide the numbers

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

Solution

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

Solve the equation