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

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

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

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

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

Solution steps

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

Treat

y

as a constant

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}

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

Simplify

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

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

x \to 0 + lim ((1 + x)^{(\frac{1}{x})})

\frac{d}{d x} (25 x^{3} - 6 x^{2} + 7 x - 88)

\frac{d y}{d x} = 0.1 y (1 - \frac{y}{500})

9 y^{^{''}} - 24 y^{^{'}} + 16 y = 0

y = \frac{1}{( x ^{2} - 2 ) ( x ^{2} + x + 5 )}

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