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

The first d/(dy)(xln(xy)) is x/y

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d/(dy)(xln(xy))

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

\frac{x}{y}

Solution steps

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

Treat

x

as a constant

Take the constant out:

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

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

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

= x \frac{1}{x y} x

Simplify

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

= \frac{x}{y}

f (x) = (2 x + 7)^{2}

- 16 t^{2}

x^{4} \frac{d y}{d x} + 2 x^{3} y = 1

\frac{d}{d x} (\frac{1}{4 - x})

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