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

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

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

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

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

Solution steps

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

Treat

y

as a constant

Apply the chain rule:

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

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

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

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

Simplify

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

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

f (x) = \frac{4 x}{x ^{2} + 1}, at x = 0

\int 6 x cos (x) d x

y = x^{2}, y = 3 x

n = 0 \sum \infty 7^{2}

\frac{d y}{d x} = sin (7 x)

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