What is (\partial)/(\partial y)(arctan(xy)) ?

The answer to (\partial)/(\partial y)(arctan(xy)) is x/(x^2y^2+1)

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(\partial)/(\partial y)(arctan(xy))

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

\frac{x}{x ^{2} y ^{2} + 1}

Solution steps

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

Treat

x

as a constant

Apply the chain rule:

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

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

\frac{\partial}{\partial y} (x y) = x

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

Simplify

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

= \frac{x}{x ^{2} y ^{2} + 1}

What is (\partial)/(\partial y)(arctan(xy)) ?
The answer to (\partial)/(\partial y)(arctan(xy)) is x/(x^2y^2+1)