What is (\partial)/(\partial y)(6ln(6x^3+y^2)) ?

The answer to (\partial)/(\partial y)(6ln(6x^3+y^2)) is (12y)/(6x^3+y^2)

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(\partial)/(\partial y)(6ln(6x^3+y^2))

\frac{\partial}{\partial y} (6 ln (6 x^{3} + y^{2}))

\frac{12 y}{6 x ^{3} + y ^{2}}

Solution steps

\frac{\partial}{\partial y} (6 ln (6 x^{3} + y^{2}))

Treat

x

as a constant

Take the constant out:

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

= 6 \frac{\partial}{\partial y} (ln (6 x^{3} + y^{2}))

Apply the chain rule:

\frac{1}{6 x ^{3} + y ^{2}} \frac{\partial}{\partial y} (6 x^{3} + y^{2})

= \frac{1}{6 x ^{3} + y ^{2}} \frac{\partial}{\partial y} (6 x^{3} + y^{2})

\frac{\partial}{\partial y} (6 x^{3} + y^{2}) = 2 y

= 6 \cdot \frac{1}{6 x ^{3} + y ^{2}} \cdot 2 y

Simplify

6 \cdot \frac{1}{6 x ^{3} + y ^{2}} \cdot 2 y : \frac{12 y}{6 x ^{3} + y ^{2}}

= \frac{12 y}{6 x ^{3} + y ^{2}}

What is (\partial)/(\partial y)(6ln(6x^3+y^2)) ?
The answer to (\partial)/(\partial y)(6ln(6x^3+y^2)) is (12y)/(6x^3+y^2)