What is (\partial)/(\partial y)(x^3ln(y^6)) ?

The answer to (\partial)/(\partial y)(x^3ln(y^6)) is (6x^3)/y

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(\partial)/(\partial y)(x^3ln(y^6))

\frac{\partial}{\partial y} (x^{3} ln (y^{6}))

\frac{6 x ^{3}}{y}

Solution steps

\frac{\partial}{\partial y} (x^{3} ln (y^{6}))

Treat

x

as a constant

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= x^{3} \frac{\partial}{\partial y} (ln (y^{6}))

Apply the chain rule:

\frac{1}{y ^{6}} \frac{\partial}{\partial y} (y^{6})

= \frac{1}{y ^{6}} \frac{\partial}{\partial y} (y^{6})

\frac{\partial}{\partial y} (y^{6}) = 6 y^{5}

= x^{3} \frac{1}{y ^{6}} \cdot 6 y^{5}

Simplify

x^{3} \frac{1}{y ^{6}} \cdot 6 y^{5} : \frac{6 x ^{3}}{y}

= \frac{6 x ^{3}}{y}

What is (\partial)/(\partial y)(x^3ln(y^6)) ?
The answer to (\partial)/(\partial y)(x^3ln(y^6)) is (6x^3)/y