What is (\partial)/(\partial y)(arctan(2xy)) ?

The answer to (\partial)/(\partial y)(arctan(2xy)) is (2x)/(4x^2y^2+1)

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(\partial)/(\partial y)(arctan(2xy))

\frac{\partial}{\partial y} (arctan (2 x y))

\frac{2 x}{4 x ^{2} y ^{2} + 1}

Solution steps

\frac{\partial}{\partial y} (arctan (2 x y))

Treat

x

as a constant

Apply the chain rule:

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

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

\frac{\partial}{\partial y} (2 x y) = 2 x

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

Simplify

\frac{1}{( 2 x y ) ^{2} + 1} \cdot 2 x : \frac{2 x}{4 x ^{2} y ^{2} + 1}

= \frac{2 x}{4 x ^{2} y ^{2} + 1}

\frac{d}{d x} (x^{4} (x^{2} - 2 x)^{2})

n = 0 \sum \infty (\frac{1}{e})^{n}

(1 - 2) (12)

\int (x^{3} - 5 x)^{5} (3 x^{2} - 5) d x

\frac{d}{d x} ((5 x)^{2})

What is (\partial)/(\partial y)(arctan(2xy)) ?
The answer to (\partial)/(\partial y)(arctan(2xy)) is (2x)/(4x^2y^2+1)