What is the d/(dy)(arctan(y/x)) ?

The d/(dy)(arctan(y/x)) is x/(y^2+x^2)

What is the first d/(dy)(arctan(y/x)) ?

The first d/(dy)(arctan(y/x)) is x/(y^2+x^2)

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\frac{d}{d y} (arctan (\frac{y}{x}))

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

Solution steps

\frac{d}{d y} (arctan (\frac{y}{x}))

Treat

x

as a constant

Apply the chain rule:

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

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

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

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

Simplify

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

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

12 (x - \frac{1}{x})^{4}

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What is the d/(dy)(arctan(y/x)) ?
The d/(dy)(arctan(y/x)) is x/(y^2+x^2)
What is the first d/(dy)(arctan(y/x)) ?
The first d/(dy)(arctan(y/x)) is x/(y^2+x^2)