Find the derivative of arctan(y/x)

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

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Answer

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

Step by step solution

Problem

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

Taking the derivative of arctangent

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

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

The derivative of the constant function is equal to zero

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

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

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

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

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

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

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

Simplifying the fraction by $x$

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

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

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

Multiplying fractions

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

Answer

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

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Problem Analysis

Main topic:

Differential calculus

Time to solve it:

0.26 seconds

Views:

113