Solve z^2 = y^2 x^2

Evaluate

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

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

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

Find the derivative

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

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

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

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

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

Multiply the terms

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

Divide both sides

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

Divide the numbers

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

Solution

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

Solve the equation