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

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

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

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

\frac{3 y}{x ^{2} + y ^{2}}

Solution steps

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

Treat

y

as a constant

Take the constant out:

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

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

Apply the chain rule:

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

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

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

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

Simplify

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

= \frac{3 y}{x ^{2} + y ^{2}}

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