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

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

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

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

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

Solution steps

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

Treat

y

as a constant

Apply the Product Rule:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = x, g = arcsin (\frac{y}{x})

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

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

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

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

Simplify

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

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

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