What is (\partial)/(\partial x)(3xarctan(x/y)) ?

The answer to (\partial)/(\partial x)(3xarctan(x/y)) is 3(arctan(x/y)+(xy)/(x^2+y^2))

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(\partial)/(\partial x)(3xarctan(x/y))

\frac{\partial}{\partial x} (3 x arctan (\frac{x}{y}))

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

Solution steps

\frac{\partial}{\partial x} (3 x arctan (\frac{x}{y}))

Treat

y

as a constant

Take the constant out:

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

= 3 \frac{\partial}{\partial x} (x arctan (\frac{x}{y}))

Apply the Product Rule:

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

f = x, g = arctan (\frac{x}{y})

= 3 (\frac{\partial}{\partial x} (x) arctan (\frac{x}{y}) + \frac{\partial}{\partial x} (arctan (\frac{x}{y})) x)

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

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

= 3 (1 \cdot arctan (\frac{x}{y}) + \frac{y}{x ^{2} + y ^{2}} x)

Simplify

3 (1 \cdot arctan (\frac{x}{y}) + \frac{y}{x ^{2} + y ^{2}} x) : 3 (arctan (\frac{x}{y}) + \frac{x y}{x ^{2} + y ^{2}})

= 3 (arctan (\frac{x}{y}) + \frac{x y}{x ^{2} + y ^{2}})

4 e^{x} + x

\frac{d y}{d x} = - \frac{2 x}{y}

(x + 1) (4 x^{2} - 4 x - 8)

\frac{\partial}{\partial x} (x^{3} y - x y^{3})

(- 12 x y^{2} - 12 y) + (- 12 x^{2} y - 12 x) y^{^{'}} = 0

What is (\partial)/(\partial x)(3xarctan(x/y)) ?
The answer to (\partial)/(\partial x)(3xarctan(x/y)) is 3(arctan(x/y)+(xy)/(x^2+y^2))