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Step-by-step Solution

Find the derivative of $arctan\left(\frac{x}{y}\right)$

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Answer

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

Step-by-step explanation

Problem to solve:

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

Taking the derivative of arctangent

$\frac{1}{1+\left(\frac{x}{y}\right)^2}\cdot\frac{d}{dx}\left(\frac{x}{y}\right)$
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 ${\displaystyle h'(x) = \frac{f'(x) \cdot g(x) - g'(x) \cdot f(x)}{g(x)^2}}$

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

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Answer

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

Main topic:

Differential calculus

Used formulas:

3. See formulas

Time to solve it:

~ 0.66 seconds

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