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

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

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

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

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

Solution steps

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

Treat

y

as a constant

Take the constant out:

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

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

Apply exponent rule:

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

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

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