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

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

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

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

\frac{1}{y ^{2} - x ^{2}}

Solution steps

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

Treat

y

as a constant

Apply the chain rule:

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

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

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

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

Simplify

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

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

x \to 0 lim (- ((1 + x^{2})^{c o t (x)}))

\frac{d}{d t} (\frac{( 2 t y )}{3})

y = e^{x}, y = 2 e^{- x} + 1, x = 0

\frac{\partial}{\partial x} (- 4 sin (2 x))

\frac{d y}{d x}, x^{3} + y^{3} = 36

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