What is (\partial)/(\partial x)(1/2 yarctan(x/y)) ?

The answer to (\partial)/(\partial x)(1/2 yarctan(x/y)) is (y^2)/(2(x^2+y^2))

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

(\partial)/(\partial x)(1/2 yarctan(x/y))

\frac{\partial}{\partial x} (\frac{1}{2} y arctan (\frac{x}{y}))

\frac{y ^{2}}{2 ( x ^{2} + y ^{2} )}

Solution steps

\frac{\partial}{\partial x} (\frac{1}{2} y arctan (\frac{x}{y}))

Treat

y

as a constant

Take the constant out:

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

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

\int \frac{x ^{2} - 3 x - 7}{( 2 x + 3 ) ( x + 1 )} d x

\int_{1}^{e^{3}} \frac{1}{x ( 1 + ln ( x ) )} d x

y^{^{''}} + 25 y = 4 csc^{2} (5 t)

\frac{\partial}{\partial x} (3 x cos (y))

x \to 0 lim (\frac{1}{cos ( 8 x )})

What is (\partial)/(\partial x)(1/2 yarctan(x/y)) ?
The answer to (\partial)/(\partial x)(1/2 yarctan(x/y)) is (y^2)/(2(x^2+y^2))