What is (\partial)/(\partial x)(8xarctan(x/y)) ?

The answer to (\partial)/(\partial x)(8xarctan(x/y)) is 8(arctan(x/y)+(xy)/(x^2+y^2))

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(\partial)/(\partial x)(8xarctan(x/y))

\frac{\partial}{\partial x} (8 x arctan (\frac{x}{y}))

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

Solution steps

\frac{\partial}{\partial x} (8 x arctan (\frac{x}{y}))

Treat

y

as a constant

Take the constant out:

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

= 8 \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})

= 8 (\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}}

= 8 (1 \cdot arctan (\frac{x}{y}) + \frac{y}{x ^{2} + y ^{2}} x)

Simplify

8 (1 \cdot arctan (\frac{x}{y}) + \frac{y}{x ^{2} + y ^{2}} x) : 8 (arctan (\frac{x}{y}) + \frac{x y}{x ^{2} + y ^{2}})

= 8 (arctan (\frac{x}{y}) + \frac{x y}{x ^{2} + y ^{2}})

What is (\partial)/(\partial x)(8xarctan(x/y)) ?
The answer to (\partial)/(\partial x)(8xarctan(x/y)) is 8(arctan(x/y)+(xy)/(x^2+y^2))