What is the d/(dy)(arctan(sqrt((x-y)/x))) ?

The d/(dy)(arctan(sqrt((x-y)/x))) is -(sqrt(x))/(2(2x-y)sqrt(x-y))

What is the first d/(dy)(arctan(sqrt((x-y)/x))) ?

The first d/(dy)(arctan(sqrt((x-y)/x))) is -(sqrt(x))/(2(2x-y)sqrt(x-y))

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\frac{d}{d y} (arctan (\frac{x - y}{x}))

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

Solution steps

\frac{d}{d y} (arctan (\frac{x - y}{x}))

Treat

x

as a constant

Apply the chain rule:

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

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

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

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

Simplify

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

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

\frac{\partial}{\partial x} (\frac{( x - y )}{c ( a - b )})

\frac{d}{d t} (8 t)

\int \frac{5}{3} d x

\int_{6}^{8} \frac{x}{x ^{2} + 6 x + 13} d x

\frac{\partial}{\partial x} (ln (8 x))

What is the d/(dy)(arctan(sqrt((x-y)/x))) ?
The d/(dy)(arctan(sqrt((x-y)/x))) is -(sqrt(x))/(2(2x-y)sqrt(x-y))
What is the first d/(dy)(arctan(sqrt((x-y)/x))) ?
The first d/(dy)(arctan(sqrt((x-y)/x))) is -(sqrt(x))/(2(2x-y)sqrt(x-y))