What is (\partial)/(\partial y)(xarcsin(y/x)) ?

The answer to (\partial)/(\partial y)(xarcsin(y/x)) is x/(sqrt(x^2-y^2))

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(\partial)/(\partial y)(xarcsin(y/x))

\frac{\partial}{\partial y} (x arcsin (\frac{y}{x}))

\frac{x}{x ^{2} - y ^{2}}

Solution steps

\frac{\partial}{\partial y} (x arcsin (\frac{y}{x}))

Treat

x

as a constant

Take the constant out:

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

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

f (x) = (x + 5) (9 x + 5)

\frac{d}{d x} ((cos (x) + 2)^{3} (2 x)^{(- 3 x)})

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

\int t e^{- t} d t

- 52 x^{2} + 2 x + 2

What is (\partial)/(\partial y)(xarcsin(y/x)) ?
The answer to (\partial)/(\partial y)(xarcsin(y/x)) is x/(sqrt(x^2-y^2))