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

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

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(\partial)/(\partial x)(arccos(y/x))

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

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

Solution steps

\frac{\partial}{\partial x} (arccos (\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}}

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