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

The d/(dy)((x+y)/(xy)) is -1/(y^2)

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

The first d/(dy)((x+y)/(xy)) is -1/(y^2)

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

- \frac{1}{y ^{2}}

Solution steps

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

Treat

x

as a constant

Take the constant out:

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

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

Apply the Quotient Rule:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

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

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

\frac{d y}{d y} = 1

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

Simplify

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

= - \frac{1}{y ^{2}}

y = sin^{4} (x)

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