Solve x: z = y^3x-1

Evaluate

x \div z = y^{3} x - 1

Rewrite the expression

\frac{x}{z} = y^{3} x - 1

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

\frac{\partial}{\partial x} (\frac{x}{z}) = \frac{\partial}{\partial x} (y^{3} x - 1)

Use differentiation rule \frac{\partial}{\partial x} (\frac{f ( x )}{g ( x )}) = \frac{\frac{\partial}{\partial x} ( f ( x )) \times g ( x ) - f ( x ) \times \frac{\partial}{\partial x} ( g ( x ) )}{( g ( x ) ) ^{2}}

\frac{\frac{\partial}{\partial x} ( x ) z - x \times \frac{\partial}{\partial x} ( z )}{z ^{2}} = \frac{\partial}{\partial x} (y^{3} x - 1)

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

\frac{1 \times z - x \times \frac{\partial}{\partial x} ( z )}{z ^{2}} = \frac{\partial}{\partial x} (y^{3} x - 1)

Evaluate

\frac{1 \times z - x \frac{\partial z}{\partial x}}{z ^{2}} = \frac{\partial}{\partial x} (y^{3} x - 1)

Any expression multiplied by 1 remains the same

\frac{z - x \frac{\partial z}{\partial x}}{z ^{2}} = \frac{\partial}{\partial x} (y^{3} 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{z - x \frac{\partial z}{\partial x}}{z ^{2}} = \frac{\partial}{\partial x} (y^{3} x) - \frac{\partial}{\partial x} (1)

Evaluate

More Steps

\frac{z - x \frac{\partial z}{\partial x}}{z ^{2}} = y^{3} - \frac{\partial}{\partial x} (1)

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

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

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

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

Cross multiply

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

Simplify the equation

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

Move the constant to the right side

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

Divide both sides

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

Divide the numbers

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

Solution

More Steps

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

Solve the equation