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

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

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

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

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

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

Solution steps

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

Apply the Quotient Rule:

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

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

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

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

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

Expand

(3 a x^{2} + 2 x) (1 + x + x^{2}) - (2 x + 1) (a x^{3} + x^{2} + 2) : a x^{4} + 2 a x^{3} + 3 a x^{2} + x^{2} - 2 x - 2

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

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