Simplify 1/(sqrt(x^2+1))-(x^2)/((x^2+1)^(3/2))

Makayla Reilly 2022-09-14 Answered
Simplify 1 x 2 + 1 x 2 ( x 2 + 1 ) 3 / 2
I want to know why
1 x 2 + 1 x 2 ( x 2 + 1 ) 3 / 2
can be simplified into
1 ( x 2 + 1 ) 3 / 2
I tried to simplify by rewriting radicals and fractions. I was hoping to see a clever trick (e.g. adding a clever zero, multiplying by a clever one? Quadratic completion?)
1 x 2 + 1 x 2 ( x 2 + 1 ) 3 / 2 = = ( x 2 + 1 ) 1 / 2 x 2 ( x 2 + 1 ) 3 / 2 = ( x 2 + 1 ) 1 / 2 ( 1 x 2 ( x 2 + 1 ) 1 ) = . . .
To give a bit more context, I was calculating the derivative of x x 2 + 1 in order to use newtons method for approximating the roots.
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Answers (2)

Manuel Leach
Answered 2022-09-15 Author has 13 answers
x is just a shorthand for x 1 / 2 . Hence we can multiply the two halves of the first fraction in the first term by x 2 + 1:
1 x 2 + 1 x 2 ( x 2 + 1 ) 3 / 2 = x 2 + 1 ( x 2 + 1 ) 3 / 2 x 2 ( x 2 + 1 ) 3 / 2
and the target expression follows.

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Gaintentavyw4
Answered 2022-09-16 Author has 1 answers
HINT: we have
1 x 2 + 1 x 2 x 2 + 1 3 = x 2 + 1 x 2 x 2 + 1 3

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