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

The answer to (\partial)/(\partial y)(arcsin(x/y)) is -x/(ysqrt(y^2-x^2))

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(\partial)/(\partial y)(arcsin(x/y))

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

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

Solution steps

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

Treat

x

as a constant

Apply the chain rule:

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

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

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

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

Simplify

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

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

e^{x^{2} + x}

\frac{d}{d x} (y^{2} - x)

e^{l n (s i n (x^{2}))} 2 x + 1

x \to 0 lim (\frac{ln ( x + 1 )}{x})

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