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

The derivative of (1+x^2+x^4/(1+x+x^2)) is (2x^5+3x^4+4x^3+x^2-1)/((1+x+x^2)^2)

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

The first derivative of (1+x^2+x^4/(1+x+x^2)) is (2x^5+3x^4+4x^3+x^2-1)/((1+x+x^2)^2)

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

\frac{2 x ^{5} + 3 x ^{4} + 4 x ^{3} + x ^{2} - 1}{( 1 + x + x ^{2} ) ^{2}}

Solution steps

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

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

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

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

Expand

(4 x^{3} + 2 x) (1 + x + x^{2}) - (2 x + 1) (1 + x^{2} + x^{4}) : 2 x^{5} + 3 x^{4} + 4 x^{3} + x^{2} - 1

= \frac{2 x ^{5} + 3 x ^{4} + 4 x ^{3} + x ^{2} - 1}{( 1 + x + x ^{2} ) ^{2}}

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

\int e^{1 - x} d x

\int_{0}^{3} \frac{1}{9 + x ^{2}} d x

\int_{- 2}^{0} x^{3} - 4 x d x

What is the derivative of (1+x^2+x^4/(1+x+x^2)) ?
The derivative of (1+x^2+x^4/(1+x+x^2)) is (2x^5+3x^4+4x^3+x^2-1)/((1+x+x^2)^2)
What is the first derivative of (1+x^2+x^4/(1+x+x^2)) ?
The first derivative of (1+x^2+x^4/(1+x+x^2)) is (2x^5+3x^4+4x^3+x^2-1)/((1+x+x^2)^2)