What is (\partial)/(\partial y)(yln(x)+1/9 y^3) ?

The answer to (\partial)/(\partial y)(yln(x)+1/9 y^3) is ln(x)+(y^2)/3

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(\partial)/(\partial y)(yln(x)+1/9 y^3)

\frac{\partial}{\partial y} (y ln (x) + \frac{1}{9} y^{3})

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

Solution steps

\frac{\partial}{\partial y} (y ln (x) + \frac{1}{9} y^{3})

Simplify

y ln (x) + \frac{1}{9} y^{3} : y ln (x) + \frac{y ^{3}}{9}

= \frac{\partial}{\partial y} (y ln (x) + \frac{y ^{3}}{9})

Treat

x

as a constant

Apply the Sum/Difference Rule:

(f \pm g)^{'} = f^{'} \pm g^{'}

= \frac{\partial}{\partial y} (y ln (x)) + \frac{\partial}{\partial y} (\frac{y ^{3}}{9})

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

\frac{\partial}{\partial y} (\frac{y ^{3}}{9}) = \frac{y ^{2}}{3}

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

x \to 0 lim (\frac{3 cos ( x )}{x})

\frac{d}{d x} (\frac{1 - sec ( x )}{tan ( x )})

f (x) = 5 x - 2 x, (1, 3)

\frac{d y}{d x}, y = x^{3} + 7

\frac{d}{d x} (- x^{3})

What is (\partial)/(\partial y)(yln(x)+1/9 y^3) ?
The answer to (\partial)/(\partial y)(yln(x)+1/9 y^3) is ln(x)+(y^2)/3