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

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

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

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

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

Solution steps

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

Treat

a, y

as a constant

Apply the Product Rule:

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

f = x^{a}, g = (\frac{y}{x})^{1 - a}

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

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

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

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

Simplify

a x^{a - 1} (\frac{y}{x})^{1 - a} + (- y^{1 - a} x^{a - 2} (1 - a)) x^{a} : a (\frac{y}{x})^{1 - a} x^{a - 1} - x^{2 a - 2} y^{1 - a} (1 - a)

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

\frac{\partial}{\partial y} (ln (y) + 2 x y^{3})

n = 0 \sum \infty n! x^{n}

\int 2 sin (t) cos (t) e^{- c o s (t)} d t

y = x^{4} + 3 e^{x}

tan (x + 1)

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