What is (\partial)/(\partial y)(sqrt((x+2)/(y+1))) ?

The answer to (\partial)/(\partial y)(sqrt((x+2)/(y+1))) is -(sqrt(x+2))/(2(y+1)^{3/2)}

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(\partial)/(\partial y)(sqrt((x+2)/(y+1)))

\frac{\partial}{\partial y} (\frac{x + 2}{y + 1})

- \frac{x + 2}{2 ( y + 1 ) ^{\frac{3}{2}}}

Solution steps

\frac{\partial}{\partial y} (\frac{x + 2}{y + 1})

Treat

x

as a constant

Apply the chain rule:

\frac{1}{2 \frac{x + 2}{y + 1}} \frac{\partial}{\partial y} (\frac{x + 2}{y + 1})

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

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

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

Simplify

\frac{1}{2 \frac{x + 2}{y + 1}} (- \frac{x + 2}{( y + 1 ) ^{2}}) : - \frac{x + 2}{2 ( y + 1 ) ^{\frac{3}{2}}}

= - \frac{x + 2}{2 ( y + 1 ) ^{\frac{3}{2}}}

\int (x + 2) (x - 3) d x

\int_{0}^{1} t 4 + 9 t^{2} d t

\int e^{-}

\frac{d}{d x} (\frac{1}{2 π} e^{\frac{- x ^{2}}{2}})

x \to 3 lim (\frac{x - 3}{x + 3})

What is (\partial)/(\partial y)(sqrt((x+2)/(y+1))) ?
The answer to (\partial)/(\partial y)(sqrt((x+2)/(y+1))) is -(sqrt(x+2))/(2(y+1)^{3/2)}