What is the derivative of (x^2(x+2^4)/((2x^2-1)^3)) ?

The derivative of (x^2(x+2^4)/((2x^2-1)^3)) is (2x(x+2)^3(-8x^2-3x-2))/((2x^2-1)^4)

What is the first derivative of (x^2(x+2^4)/((2x^2-1)^3)) ?

The first derivative of (x^2(x+2^4)/((2x^2-1)^3)) is (2x(x+2)^3(-8x^2-3x-2))/((2x^2-1)^4)

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\frac{d}{d x} (\frac{x ^{2} ( x + 2 ) ^{4}}{( 2 x ^{2} - 1 ) ^{3}})

\frac{2 x ( x + 2 ) ^{3} ( - 8 x ^{2} - 3 x - 2 )}{( 2 x ^{2} - 1 ) ^{4}}

Solution steps

\frac{d}{d x} (\frac{x ^{2} ( x + 2 ) ^{4}}{( 2 x ^{2} - 1 ) ^{3}})

Apply the Quotient Rule:

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

= \frac{\frac{d}{d x} ( x ^{2} ( x + 2 ) ^{4} ) ( 2 x ^{2} - 1 ) ^{3} - \frac{d}{d x} ( ( 2 x ^{2} - 1 ) ^{3} ) x ^{2} ( x + 2 ) ^{4}}{( ( 2 x ^{2} - 1 ) ^{3} ) ^{2}}

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

\frac{d}{d x} ((2 x^{2} - 1)^{3}) = 12 x (2 x^{2} - 1)^{2}

= \frac{( 2 x ( x + 2 ) ^{4} + 4 ( x + 2 ) ^{3} x ^{2} ) ( 2 x ^{2} - 1 ) ^{3} - 12 x ( 2 x ^{2} - 1 ) ^{2} x ^{2} ( x + 2 ) ^{4}}{( ( 2 x ^{2} - 1 ) ^{3} ) ^{2}}

Simplify

\frac{( 2 x ( x + 2 ) ^{4} + 4 ( x + 2 ) ^{3} x ^{2} ) ( 2 x ^{2} - 1 ) ^{3} - 12 x ( 2 x ^{2} - 1 ) ^{2} x ^{2} ( x + 2 ) ^{4}}{( ( 2 x ^{2} - 1 ) ^{3} ) ^{2}} : \frac{2 x ( x + 2 ) ^{3} ( - 8 x ^{2} - 3 x - 2 )}{( 2 x ^{2} - 1 ) ^{4}}

= \frac{2 x ( x + 2 ) ^{3} ( - 8 x ^{2} - 3 x - 2 )}{( 2 x ^{2} - 1 ) ^{4}}

What is the derivative of (x^2(x+2^4)/((2x^2-1)^3)) ?
The derivative of (x^2(x+2^4)/((2x^2-1)^3)) is (2x(x+2)^3(-8x^2-3x-2))/((2x^2-1)^4)
What is the first derivative of (x^2(x+2^4)/((2x^2-1)^3)) ?
The first derivative of (x^2(x+2^4)/((2x^2-1)^3)) is (2x(x+2)^3(-8x^2-3x-2))/((2x^2-1)^4)