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

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

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

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

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

Solution steps

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

Treat

x

as a constant

Take the constant out:

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

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

Apply exponent rule:

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

= e^{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

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

Simplify

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

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

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