What is (\partial)/(\partial y)(xarctan(x+2y)) ?

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

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(\partial)/(\partial y)(xarctan(x+2y))

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

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

Solution steps

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

Treat

x

as a constant

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

e^{x} - e^{- x} - 2 x

\int 2 x (x^{2} + 1)^{- 6} d x

f (t) = \frac{1}{( t ^{2} + 3 t + 1 ) ^{\frac{5}{2}}}

\int 1 - e^{x} d x

x \to - \infty lim (x^{3} - x)

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