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

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

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

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

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

Solution steps

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

Treat

y

as a constant

Take the constant out:

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

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

Apply exponent rule:

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

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

\frac{\partial}{\partial x} (x^{3} + y^{3})

x^{2} sin (x)

\frac{d}{d x} ((x - cos (x)) sin (x))

4 x^{3} - 5 x + 3

x \to 0 lim ((\frac{1}{x}) sin (x))

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