What is (\partial)/(\partial x)(zarctan(x/y)) ?

The answer to (\partial)/(\partial x)(zarctan(x/y)) is (zy)/(x^2+y^2)

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

(\partial)/(\partial x)(zarctan(x/y))

\frac{\partial}{\partial x} (z arctan (\frac{x}{y}))

\frac{zy}{x ^{2} + y ^{2}}

Solution steps

\frac{\partial}{\partial x} (z arctan (\frac{x}{y}))

Treat

z, y

as constants

Take the constant out:

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

= z \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}

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

Simplify

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

= \frac{zy}{x ^{2} + y ^{2}}

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