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

The first d/(dy)(yln(y)) is ln(y)+1

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d/(dy)(yln(y))

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

ln (y) + 1

Solution steps

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

Apply the Product Rule:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = y, g = ln (y)

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

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

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

= 1 \cdot ln (y) + \frac{1}{y} y

Simplify

1 \cdot ln (y) + \frac{1}{y} y : ln (y) + 1

= ln (y) + 1