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

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

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

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

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

Solution steps

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

Treat

y

as a constant

Apply the Product Rule:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = x, g = arctan (x + 2 y)

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

\frac{\partial}{\partial x} (x) = 1

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

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

Simplify

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

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

\frac{d}{d x} (\frac{2 cos ( x )}{x + 1})

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

\frac{d}{d x} (\frac{10 x}{x ^{2} - 6})

f (x) = x^{2} - 5, (- 3, 4)

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

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