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

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

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

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

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

Solution steps

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

Treat

x

as a constant

Take the constant out:

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

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

Apply exponent rule:

\frac{1}{a} = a^{- 1}

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

y^{^{''}} - 10 y^{^{'}} + 2 y = 0

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

4 sin (π t)

\int x^{3} - 3 x^{- 4} + 5 d x

\frac{d}{d x} (xx)

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