What is the d/(dy)(x/(x+y)) ?

The d/(dy)(x/(x+y)) is -x/((x+y)^2)

What is the first d/(dy)(x/(x+y)) ?

The first d/(dy)(x/(x+y)) is -x/((x+y)^2)

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

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

Solution steps

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

Treat

x

as a constant

Take the constant out:

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

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

Apply exponent rule:

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

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

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\int_{0}^{4} (5 - t) t d t