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

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

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

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

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

Solution steps

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

Treat

x

as a constant

Take the constant out:

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

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

Apply exponent rule:

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

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

\frac{d}{d x} ((x^{2} + 3)^{s i n (x)})

e^{3 t} (9 - 4 t + 10 sin (\frac{t}{2}))

x \to 0 + lim (x ln (e^{x} - 1))

f (x) = \frac{x ^{2} - 99}{x - 10}

f (x) = e^{5 x^{2}}

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