What is the d/(dy)(y^y) ?

The d/(dy)(y^y) is y^y(ln(y)+1)

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

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

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

y^{y} (ln (y) + 1)

Solution steps

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

Apply exponent rule:

a^{b} = e^{b l n (a)}

y^{y} = e^{y l n (y)}

= \frac{d}{d y} (e^{y l n (y)})

Apply the chain rule:

e^{y l n (y)} \frac{d}{d y} (y ln (y))

= e^{y l n (y)} \frac{d}{d y} (y ln (y))

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

= e^{y l n (y)} (ln (y) + 1)

e^{y l n (y)} = y^{y}

= y^{y} (ln (y) + 1)