What is (\partial)/(\partial x)(arcsin(y/((x^2+y^2)^{1/2)})) ?

The answer to (\partial)/(\partial x)(arcsin(y/((x^2+y^2)^{1/2)})) is -y/(x^2+y^2)

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

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

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

Solution steps

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

Treat

y

as a constant

Apply the chain rule:

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

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

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

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

Simplify

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

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

\int tan (x) - 2 sin (x) d x

\frac{d}{d x} (\frac{300}{x})

\frac{\partial}{\partial x} (x yz)

f (x) g (x) h (x)

What is (\partial)/(\partial x)(arcsin(y/((x^2+y^2)^{1/2)})) ?
The answer to (\partial)/(\partial x)(arcsin(y/((x^2+y^2)^{1/2)})) is -y/(x^2+y^2)