Solve x^3 y^3 z^3=33

Evaluate

x^{3} y^{3} z^{3} = 33

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

\frac{\partial}{\partial x} (x^{3} y^{3} z^{3}) = \frac{\partial}{\partial x} (33)

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^{3} y^{3}) z^{3} + x^{3} y^{3} \times \frac{\partial}{\partial x} (z^{3}) = \frac{\partial}{\partial x} (33)

Evaluate

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3 x^{2} y^{3} z^{3} + x^{3} y^{3} \times \frac{\partial}{\partial x} (z^{3}) = \frac{\partial}{\partial x} (33)

Evaluate

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3 x^{2} y^{3} z^{3} + x^{3} y^{3} \times 3 z^{2} \frac{\partial z}{\partial x} = \frac{\partial}{\partial x} (33)

Evaluate

3 x^{2} y^{3} z^{3} + 3 z^{2} x^{3} y^{3} \frac{\partial z}{\partial x} = \frac{\partial}{\partial x} (33)

Find the partial derivative

3 x^{2} y^{3} z^{3} + 3 z^{2} x^{3} y^{3} \frac{\partial z}{\partial x} = 0

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

3 z^{2} x^{3} y^{3} \frac{\partial z}{\partial x} = 0 - 3 x^{2} y^{3} z^{3}

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

3 z^{2} x^{3} y^{3} \frac{\partial z}{\partial x} = - 3 x^{2} y^{3} z^{3}

Divide both sides

\frac{3 z ^{2} x ^{3} y ^{3} \frac{\partial z}{\partial x}}{3 z ^{2} x ^{3} y ^{3}} = \frac{- 3 x ^{2} y ^{3} z ^{3}}{3 z ^{2} x ^{3} y ^{3}}

Divide the numbers

\frac{\partial z}{\partial x} = \frac{- 3 x ^{2} y ^{3} z ^{3}}{3 z ^{2} x ^{3} y ^{3}}

Solution

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

Solve the equation