What is (\partial)/(\partial x)((6x)/(3x^2+2y^2+9)) ?

The answer to (\partial)/(\partial x)((6x)/(3x^2+2y^2+9)) is (6(-3x^2+2y^2+9))/((3x^2+2y^2+9)^2)

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(\partial)/(\partial x)((6x)/(3x^2+2y^2+9))

\frac{\partial}{\partial x} (\frac{6 x}{3 x ^{2} + 2 y ^{2} + 9})

\frac{6 ( - 3 x ^{2} + 2 y ^{2} + 9 )}{( 3 x ^{2} + 2 y ^{2} + 9 ) ^{2}}

Solution steps

\frac{\partial}{\partial x} (\frac{6 x}{3 x ^{2} + 2 y ^{2} + 9})

Treat

y

as a constant

Take the constant out:

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

= 6 \frac{\partial}{\partial x} (\frac{x}{3 x ^{2} + 2 y ^{2} + 9})

Apply the Quotient Rule:

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

= 6 \cdot \frac{\frac{\partial}{\partial x} ( x ) ( 3 x ^{2} + 2 y ^{2} + 9 ) - \frac{\partial}{\partial x} ( 3 x ^{2} + 2 y ^{2} + 9 ) x}{( 3 x ^{2} + 2 y ^{2} + 9 ) ^{2}}

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

\frac{\partial}{\partial x} (3 x^{2} + 2 y^{2} + 9) = 6 x

= 6 \cdot \frac{1 \cdot ( 3 x ^{2} + 2 y ^{2} + 9 ) - 6 xx}{( 3 x ^{2} + 2 y ^{2} + 9 ) ^{2}}

Simplify

6 \cdot \frac{1 \cdot ( 3 x ^{2} + 2 y ^{2} + 9 ) - 6 xx}{( 3 x ^{2} + 2 y ^{2} + 9 ) ^{2}} : \frac{6 ( - 3 x ^{2} + 2 y ^{2} + 9 )}{( 3 x ^{2} + 2 y ^{2} + 9 ) ^{2}}

= \frac{6 ( - 3 x ^{2} + 2 y ^{2} + 9 )}{( 3 x ^{2} + 2 y ^{2} + 9 ) ^{2}}

\int_{1}^{\infty} 4 e^{- 2 x} d x

\frac{d}{d x} (3 x^{2} + 4 x)

\frac{d}{d x} ((x - 1) \cdot e^{x^{2} - 4})

\int x ln (a x) d x

\frac{d}{d x} (cos (x) cos (y))

What is (\partial)/(\partial x)((6x)/(3x^2+2y^2+9)) ?
The answer to (\partial)/(\partial x)((6x)/(3x^2+2y^2+9)) is (6(-3x^2+2y^2+9))/((3x^2+2y^2+9)^2)