What is the first d/(dy)(ln(xy)) ?

The first d/(dy)(ln(xy)) is 1/y

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d/(dy)(ln(xy))

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

\frac{1}{y}

Solution steps

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

Treat

x

as a constant

Apply the chain rule:

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

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

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

= \frac{1}{x y} x

Simplify

\frac{1}{x y} x : \frac{1}{y}

= \frac{1}{y}

\frac{\partial}{\partial x} (5 cos (x))

\int 2 x (x^{2} + 1)^{- 5} d x

\frac{\partial}{\partial x} (\frac{a x}{y ^{2}})

\int \frac{( x + 4 )}{x ^{3} + 3 x ^{2} + 2 x} d x

t^{2} y^{^{''}} + 5 t y^{^{'}} + 4 y = 0