# Derive the function arctan(x/y) with respect to x

## \frac{d}{dx}\left(arctan\left(\frac{x}{y}\right)\right)

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ln
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asin
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asinh
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$\frac{1}{y\left(\frac{x^2}{y^2}+1\right)}$

## Step by step solution

Problem

$\frac{d}{dx}\left(arctan\left(\frac{x}{y}\right)\right)$
1

Taking the derivative of arctangent

$\frac{1}{\left(\frac{x}{y}\right)^2+1}\cdot\frac{d}{dx}\left(\frac{x}{y}\right)$
2

Applying the quotient rule which states that if $f(x)$ and $g(x)$ are functions and $h(x)$ is the function defined by ${\displaystyle h(x) = \frac{f(x)}{g(x)}}$, where ${g(x) \neq 0}$, then ${\displaystyle h'(x) = \frac{f'(x) \cdot g(x) - g'(x) \cdot f(x)}{g(x)^2}}$

$\frac{1}{\left(\frac{x}{y}\right)^2+1}\cdot\frac{y\frac{d}{dx}\left(x\right)-x\frac{d}{dx}\left(y\right)}{y^2}$
3

The derivative of the constant function is equal to zero

$\frac{1}{\left(\frac{x}{y}\right)^2+1}\cdot\frac{0\left(-1\right)x+y\frac{d}{dx}\left(x\right)}{y^2}$
4

Any expression multiplied by $0$ is equal to $0$

$\frac{1}{\left(\frac{x}{y}\right)^2+1}\cdot\frac{0+y\frac{d}{dx}\left(x\right)}{y^2}$
5

The derivative of the linear function is equal to $1$

$\frac{0+1y}{y^2}\cdot\frac{1}{\left(\frac{x}{y}\right)^2+1}$
6

$x+0=x$, where $x$ is any expression

$\frac{y}{y^2}\cdot\frac{1}{\left(\frac{x}{y}\right)^2+1}$
7

Simplifying the fraction by $y$

$\frac{1}{y}\cdot\frac{1}{\left(\frac{x}{y}\right)^2+1}$
8

The power of a quotient is equal to the quotient of the power of the numerator and denominator: $\displaystyle\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}$

$\frac{1}{y}\cdot\frac{1}{\frac{x^2}{y^2}+1}$
9

Multiplying fractions

$\frac{1}{y\left(\frac{x^2}{y^2}+1\right)}$

$\frac{1}{y\left(\frac{x^2}{y^2}+1\right)}$

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### Main topic:

Differential calculus

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