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

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

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

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

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

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

Solution steps

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

Treat

x

as a constant

Apply the chain rule:

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

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

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

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

Simplify

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

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

\frac{d}{d x} (\frac{x}{( 1 + 4 x ) ^{2}})

\int \frac{12}{4 x ^{2} + 5 x + 1} d x

\int (x + 3)^{2} (3 - x)^{7} d x

x \to 9 lim (x)

f (x) = x^{2} - x - 2

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