What is (\partial)/(\partial x)((z)(x^3-y^2)^{-1}) ?

The answer to (\partial)/(\partial x)((z)(x^3-y^2)^{-1}) is -(3zx^2)/((x^3-y^2)^2)

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

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

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

Solution steps

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

Treat

z, y

as constants

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

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

Apply the chain rule:

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

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

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

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

Simplify

z (- \frac{1}{( x ^{3} - y ^{2} ) ^{2}} \cdot 3 x^{2}) : - \frac{3 z x ^{2}}{( x ^{3} - y ^{2} ) ^{2}}

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

e^{- \frac{11}{10}}

e^{- w} (- w - 1)

x^{- 2 = \frac{1}{9}}

What is (\partial)/(\partial x)((z)(x^3-y^2)^{-1}) ?
The answer to (\partial)/(\partial x)((z)(x^3-y^2)^{-1}) is -(3zx^2)/((x^3-y^2)^2)