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

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

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

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

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

Solution steps

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

Treat

x

as a constant

Apply the chain rule:

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

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

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

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

Simplify

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

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

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