What is the d/(du)(arctan(u/v)) ?

The d/(du)(arctan(u/v)) is v/(u^2+v^2)

What is the first d/(du)(arctan(u/v)) ?

The first d/(du)(arctan(u/v)) is v/(u^2+v^2)

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\frac{d}{d u} (arctan (\frac{u}{v}))

\frac{v}{u ^{2} + v ^{2}}

Solution steps

\frac{d}{d u} (arctan (\frac{u}{v}))

Treat

v

as a constant

Apply the chain rule:

\frac{1}{( \frac{u}{v} ) ^{2} + 1} \frac{d}{d u} (\frac{u}{v})

= \frac{1}{( \frac{u}{v} ) ^{2} + 1} \frac{d}{d u} (\frac{u}{v})

\frac{d}{d u} (\frac{u}{v}) = \frac{1}{v}

= \frac{1}{( \frac{u}{v} ) ^{2} + 1} \cdot \frac{1}{v}

Simplify

\frac{1}{( \frac{u}{v} ) ^{2} + 1} \cdot \frac{1}{v} : \frac{v}{u ^{2} + v ^{2}}

= \frac{v}{u ^{2} + v ^{2}}

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What is the d/(du)(arctan(u/v)) ?
The d/(du)(arctan(u/v)) is v/(u^2+v^2)
What is the first d/(du)(arctan(u/v)) ?
The first d/(du)(arctan(u/v)) is v/(u^2+v^2)