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

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

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

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

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

Solution steps

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

Treat

y

as a constant

Apply the chain rule:

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

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

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

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

Simplify

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

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

\int \frac{( e ^{x} - e ^{- x} )}{e ^{2 x}} d x

\frac{\partial}{\partial s} (s cos (t))

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

x \to \infty lim (e^{5 x l n (1 + \frac{1}{x})})

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