What is (\partial)/(\partial r)(arctan(rs)) ?

The answer to (\partial)/(\partial r)(arctan(rs)) is s/(r^2s^2+1)

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(\partial)/(\partial r)(arctan(rs))

\frac{\partial}{\partial r} (arctan (rs))

\frac{s}{r ^{2} s ^{2} + 1}

Solution steps

\frac{\partial}{\partial r} (arctan (rs))

Treat

s

as a constant

Apply the chain rule:

\frac{1}{( rs ) ^{2} + 1} \frac{\partial}{\partial r} (rs)

= \frac{1}{( rs ) ^{2} + 1} \frac{\partial}{\partial r} (rs)

\frac{\partial}{\partial r} (rs) = s

= \frac{1}{( rs ) ^{2} + 1} s

Simplify

\frac{1}{( rs ) ^{2} + 1} s : \frac{s}{r ^{2} s ^{2} + 1}

= \frac{s}{r ^{2} s ^{2} + 1}

What is (\partial)/(\partial r)(arctan(rs)) ?
The answer to (\partial)/(\partial r)(arctan(rs)) is s/(r^2s^2+1)