What is (\partial)/(\partial y)(yln(x^2+y^3+5)) ?

The answer to (\partial)/(\partial y)(yln(x^2+y^3+5)) is ln(x^2+y^3+5)+(3y^3)/(x^2+y^3+5)

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(\partial)/(\partial y)(yln(x^2+y^3+5))

\frac{\partial}{\partial y} (y ln (x^{2} + y^{3} + 5))

ln (x^{2} + y^{3} + 5) + \frac{3 y ^{3}}{x ^{2} + y ^{3} + 5}

Solution steps

\frac{\partial}{\partial y} (y ln (x^{2} + y^{3} + 5))

Treat

x

as a constant

Apply the Product Rule:

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

f = y, g = ln (x^{2} + y^{3} + 5)

= \frac{\partial}{\partial y} (y) ln (x^{2} + y^{3} + 5) + \frac{\partial}{\partial y} (ln (x^{2} + y^{3} + 5)) y

\frac{\partial}{\partial y} (y) = 1

\frac{\partial}{\partial y} (ln (x^{2} + y^{3} + 5)) = \frac{3 y ^{2}}{x ^{2} + y ^{3} + 5}

= 1 \cdot ln (x^{2} + y^{3} + 5) + \frac{3 y ^{2}}{x ^{2} + y ^{3} + 5} y

Simplify

1 \cdot ln (x^{2} + y^{3} + 5) + \frac{3 y ^{2}}{x ^{2} + y ^{3} + 5} y : ln (x^{2} + y^{3} + 5) + \frac{3 y ^{3}}{x ^{2} + y ^{3} + 5}

= ln (x^{2} + y^{3} + 5) + \frac{3 y ^{3}}{x ^{2} + y ^{3} + 5}

What is (\partial)/(\partial y)(yln(x^2+y^3+5)) ?
The answer to (\partial)/(\partial y)(yln(x^2+y^3+5)) is ln(x^2+y^3+5)+(3y^3)/(x^2+y^3+5)