What is (\partial)/(\partial x)((x^αy^β)/(y^{1-α)x^{1-β}}) ?

The answer to (\partial)/(\partial x)((x^αy^β)/(y^{1-α)x^{1-β}}) is x^{α+β-2}y^{β-1+α}(α-1+β)

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(\partial)/(\partial x)((x^αy^β)/(y^{1-α)x^{1-β}})

\frac{\partial}{\partial x} (\frac{x ^{α} y ^{β}}{y ^{1 - α} x ^{1 - β}})

x^{α + β - 2} y^{β - 1 + α} (α - 1 + β)

Solution steps

\frac{\partial}{\partial x} (\frac{x ^{α} y ^{β}}{y ^{1 - α} x ^{1 - β}})

Simplify

\frac{x ^{α} y ^{β}}{y ^{1 - α} x ^{1 - β}} : x^{α - 1 + β} y^{β - 1 + α}

= \frac{\partial}{\partial x} (x^{α - 1 + β} y^{β - 1 + α})

Treat

α, β, y

as constants

Take the constant out:

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

= y^{β - 1 + α} \frac{\partial}{\partial x} (x^{α - 1 + β})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= y^{β - 1 + α} (α - 1 + β) x^{α - 1 + β - 1}

Simplify

= x^{α + β - 2} y^{β - 1 + α} (α - 1 + β)

What is (\partial)/(\partial x)((x^αy^β)/(y^{1-α)x^{1-β}}) ?
The answer to (\partial)/(\partial x)((x^αy^β)/(y^{1-α)x^{1-β}}) is x^{α+β-2}y^{β-1+α}(α-1+β)