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

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

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d/(dy)(x/y)

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

- \frac{x}{y ^{2}}

Solution steps

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

Treat

x

as a constant

Take the constant out:

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

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

Apply exponent rule:

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

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

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

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

Simplify

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

= - \frac{x}{y ^{2}}

x^{2} - 7 x - 24, - x^{2} + 3 x + 4

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

3 ln (5 x)

x^{2} + 2 x y + x \frac{d y}{d x} = 0