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

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

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

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

\frac{y}{x}

Solution steps

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

Treat

y

as a constant

Apply the Sum/Difference Rule:

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

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

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

\frac{\partial}{\partial x} (\frac{y ^{3}}{9}) = 0

= \frac{y}{x} + 0

Simplify

= \frac{y}{x}

\frac{\partial}{\partial x} (3 x^{2})

\int \frac{( x ^{2} + 2 x + 1 )}{x} d x

\int \frac{e ^{2 x}}{9 + e ^{4 x}} d x

\int \frac{1}{x \cdot ( x ^{2} - 1 ) ^{\frac{1}{2}}} d x

\frac{d}{d x} (3 x^{3} - \frac{1}{x ^{3}})

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