What is the derivative of x^ay^{1-a} ?

The derivative of x^ay^{1-a} is ax^{a-1}y^{1-a}+y^{-a}(1-a)(dy)/(dx)x^a

What is the first derivative of x^ay^{1-a} ?

The first derivative of x^ay^{1-a} is ax^{a-1}y^{1-a}+y^{-a}(1-a)(dy)/(dx)x^a

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

a x^{a - 1} y^{1 - a} + y^{- a} (1 - a) \frac{d y}{d x} x^{a}

Solution steps

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

Apply the Product Rule:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = x^{a}, g = y^{1 - a}

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

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

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

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

What is the derivative of x^ay^{1-a} ?
The derivative of x^ay^{1-a} is ax^{a-1}y^{1-a}+y^{-a}(1-a)(dy)/(dx)x^a
What is the first derivative of x^ay^{1-a} ?
The first derivative of x^ay^{1-a} is ax^{a-1}y^{1-a}+y^{-a}(1-a)(dy)/(dx)x^a