What is the derivative of (2(-x^2-4)/((x^2-4)^2)) ?

The derivative of (2(-x^2-4)/((x^2-4)^2)) is -(4x(-x^2-12))/((x^2-4)^3)

What is the first derivative of (2(-x^2-4)/((x^2-4)^2)) ?

The first derivative of (2(-x^2-4)/((x^2-4)^2)) is -(4x(-x^2-12))/((x^2-4)^3)

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

- \frac{4 x ( - x ^{2} - 12 )}{( x ^{2} - 4 ) ^{3}}

Solution steps

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

Take the constant out:

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

= 2 \frac{d}{d x} (\frac{- x ^{2} - 4}{( x ^{2} - 4 ) ^{2}})

Apply the Quotient Rule:

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

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

\frac{d}{d x} (- x^{2} - 4) = - 2 x

\frac{d}{d x} ((x^{2} - 4)^{2}) = 4 x (x^{2} - 4)

= 2 \cdot \frac{( - 2 x ) ( x ^{2} - 4 ) ^{2} - 4 x ( x ^{2} - 4 ) ( - x ^{2} - 4 )}{( ( x ^{2} - 4 ) ^{2} ) ^{2}}

Simplify

2 \cdot \frac{( - 2 x ) ( x ^{2} - 4 ) ^{2} - 4 x ( x ^{2} - 4 ) ( - x ^{2} - 4 )}{( ( x ^{2} - 4 ) ^{2} ) ^{2}} : - \frac{4 x ( - x ^{2} - 12 )}{( x ^{2} - 4 ) ^{3}}

= - \frac{4 x ( - x ^{2} - 12 )}{( x ^{2} - 4 ) ^{3}}

x \to 0 lim (\frac{tan ( 4 x )}{3 x})

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What is the derivative of (2(-x^2-4)/((x^2-4)^2)) ?
The derivative of (2(-x^2-4)/((x^2-4)^2)) is -(4x(-x^2-12))/((x^2-4)^3)
What is the first derivative of (2(-x^2-4)/((x^2-4)^2)) ?
The first derivative of (2(-x^2-4)/((x^2-4)^2)) is -(4x(-x^2-12))/((x^2-4)^3)