What is (\partial)/(\partial x)(x/(3+6x^2+7y^2)) ?

The answer to (\partial)/(\partial x)(x/(3+6x^2+7y^2)) is (-6x^2+7y^2+3)/((3+6x^2+7y^2)^2)

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(\partial)/(\partial x)(x/(3+6x^2+7y^2))

\frac{\partial}{\partial x} (\frac{x}{3 + 6 x ^{2} + 7 y ^{2}})

\frac{- 6 x ^{2} + 7 y ^{2} + 3}{( 3 + 6 x ^{2} + 7 y ^{2} ) ^{2}}

Solution steps

\frac{\partial}{\partial x} (\frac{x}{3 + 6 x ^{2} + 7 y ^{2}})

Treat

y

as a constant

Apply the Quotient Rule:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

= \frac{\frac{\partial}{\partial x} ( x ) ( 3 + 6 x ^{2} + 7 y ^{2} ) - \frac{\partial}{\partial x} ( 3 + 6 x ^{2} + 7 y ^{2} ) x}{( 3 + 6 x ^{2} + 7 y ^{2} ) ^{2}}

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

\frac{\partial}{\partial x} (3 + 6 x^{2} + 7 y^{2}) = 12 x

= \frac{1 \cdot ( 3 + 6 x ^{2} + 7 y ^{2} ) - 12 xx}{( 3 + 6 x ^{2} + 7 y ^{2} ) ^{2}}

1 \cdot (3 + 6 x^{2} + 7 y^{2}) - 12 xx = - 6 x^{2} + 7 y^{2} + 3

= \frac{- 6 x ^{2} + 7 y ^{2} + 3}{( 3 + 6 x ^{2} + 7 y ^{2} ) ^{2}}

What is (\partial)/(\partial x)(x/(3+6x^2+7y^2)) ?
The answer to (\partial)/(\partial x)(x/(3+6x^2+7y^2)) is (-6x^2+7y^2+3)/((3+6x^2+7y^2)^2)