Differentiate this

Calculus Level 2

Find the derivative of x 4 + x 2 + 1 x 2 + x + 1 \large \frac{x^4+x^2+1}{x^2+x+1}

If the answer is of the form a x + b ax+b , where a , b a,b are integers, find a + b a+b .


The answer is 1.

Vilakshan Gupta by Vilakshan Gupta , 19, India

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1 solution

Steven Yuan
Jan 28, 2018

x 4 + x 2 + 1 = x 4 + 2 x 2 + 1 − x 2 = ( x 2 + 1 ) 2 − x 2 = ( x 2 + x + 1 ) ( x 2 − x + 1 ) . \begin{aligned} x^4 + x^2 + 1 &= x^4 + 2x^2 + 1 - x^2 \\ &= (x^2 + 1)^2 - x^2 \\ &= (x^2 + x + 1)(x^2 - x + 1). \end{aligned}

Thus,

d d x [ x 4 + x 2 + 1 x 2 + x + 1 ] = d d x [ x 2 − x + 1 ] = 2 x − 1 , \dfrac{d}{dx} \left [ \dfrac{x^4 + x^2 + 1}{x^2 + x + 1} \right ] = \dfrac{d}{dx} [x^2 - x + 1] = 2x - 1,

so a + b = 2 − 1 = 1 . a + b = 2 - 1 = \boxed{1}.

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Nice Solution :)

Vilakshan Gupta - 3 years, 4 months ago

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Thanks ⌣ ¨ \ddot \smile

Steven Yuan - 3 years, 4 months ago

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