What is (\partial)/(\partial x)((12x)/(6x^2+y^2+4)) ?

The answer to (\partial)/(\partial x)((12x)/(6x^2+y^2+4)) is (12(-6x^2+y^2+4))/((6x^2+y^2+4)^2)

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(\partial)/(\partial x)((12x)/(6x^2+y^2+4))

\frac{\partial}{\partial x} (\frac{12 x}{6 x ^{2} + y ^{2} + 4})

\frac{12 ( - 6 x ^{2} + y ^{2} + 4 )}{( 6 x ^{2} + y ^{2} + 4 ) ^{2}}

Solution steps

\frac{\partial}{\partial x} (\frac{12 x}{6 x ^{2} + y ^{2} + 4})

Treat

y

as a constant

Take the constant out:

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

= 12 \frac{\partial}{\partial x} (\frac{x}{6 x ^{2} + y ^{2} + 4})

Apply the Quotient Rule:

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

= 12 \cdot \frac{\frac{\partial}{\partial x} ( x ) ( 6 x ^{2} + y ^{2} + 4 ) - \frac{\partial}{\partial x} ( 6 x ^{2} + y ^{2} + 4 ) x}{( 6 x ^{2} + y ^{2} + 4 ) ^{2}}

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

\frac{\partial}{\partial x} (6 x^{2} + y^{2} + 4) = 12 x

= 12 \cdot \frac{1 \cdot ( 6 x ^{2} + y ^{2} + 4 ) - 12 xx}{( 6 x ^{2} + y ^{2} + 4 ) ^{2}}

Simplify

12 \cdot \frac{1 \cdot ( 6 x ^{2} + y ^{2} + 4 ) - 12 xx}{( 6 x ^{2} + y ^{2} + 4 ) ^{2}} : \frac{12 ( - 6 x ^{2} + y ^{2} + 4 )}{( 6 x ^{2} + y ^{2} + 4 ) ^{2}}

= \frac{12 ( - 6 x ^{2} + y ^{2} + 4 )}{( 6 x ^{2} + y ^{2} + 4 ) ^{2}}

What is (\partial)/(\partial x)((12x)/(6x^2+y^2+4)) ?
The answer to (\partial)/(\partial x)((12x)/(6x^2+y^2+4)) is (12(-6x^2+y^2+4))/((6x^2+y^2+4)^2)