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

The answer to (\partial)/(\partial y)(zarcsin(y/x)) is z/(sqrt(x^2-y^2))

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(\partial)/(\partial y)(zarcsin(y/x))

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

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

Solution steps

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

Treat

z, x

as constants

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

\int (2 - t) e^{- s t} d t

x \to 0 lim (\frac{ln ( 1 - 2 x )}{x})

\int \frac{2 u}{1 - u ^{2}} d u

(- 9, 5), (- 3, 3)

\frac{d}{d x} (3^{x - 2})

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