What is (\partial)/(\partial x)(x/(sqrt(y-x))) ?

The answer to (\partial)/(\partial x)(x/(sqrt(y-x))) is (2y-x)/(2(y-x)sqrt(y-x))

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(\partial)/(\partial x)(x/(sqrt(y-x)))

\frac{\partial}{\partial x} (\frac{x}{y - x})

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

Solution steps

\frac{\partial}{\partial x} (\frac{x}{y - x})

Treat

y

as a constant

Apply the Quotient Rule:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

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

\frac{\partial}{\partial x} (x) = 1

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

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

Simplify

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

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

\frac{\partial}{\partial x} (x + yz)

\int_{1}^{2} \frac{13}{x ^{3} + 2 x} d x

\int \frac{2 x + 1}{x} d x

\frac{\partial}{\partial x} (e^{x y} cos (x^{2}))

x y^{2} d y = (y^{3} - x^{3}) d x

What is (\partial)/(\partial x)(x/(sqrt(y-x))) ?
The answer to (\partial)/(\partial x)(x/(sqrt(y-x))) is (2y-x)/(2(y-x)sqrt(y-x))