# Thread: [SOLVED] How to differentiate this?

1. ## [SOLVED] How to differentiate this?

$\displaystyle \frac{1-x^2}{\sqrt{1 + 2x}}$

This is my take on the question:

$\displaystyle =\frac{\sqrt{1+2x}(-2x) - (1-x^2)\frac{1}{\sqrt{1 + 2x}}}{1 + 2x}$

$\displaystyle =\frac{(-2x)(1 + 2x) - (1 - x^2)}{\sqrt{1 + 2x}} \times \frac{1}{1 + 2x}$

$\displaystyle = \frac{-2x - 1 + x^2}{\sqrt{1 + 2x}}$

$\displaystyle \frac{- (3x^2 + 2x + 1)}{(\sqrt{1 + 2x})^3}$

Can you tell me where I have gone wrong? Thanks.

2. ## Mistake

Good night(or morning or afternoon, depending on your location)

By doing this, you change the expression completely.

What you should have done is multiply the numerator and denominator by $\displaystyle \sqrt{1+2x}$, which is effectively multiplying by $\displaystyle 1$.

To see it in action
From:

$\displaystyle *(\frac{\sqrt{1+2x}}{\sqrt{1+2x}})$

$\displaystyle =\frac{-2x-4x^2-1+x^2}{\sqrt{{1+2x}}^3}$

And there you have it

from
$\displaystyle =\frac{\sqrt{1+2x}(-2x) - (1-x^2)\frac{1}{\sqrt{1 + 2x}}}{1 + 2x}$

I equated the denominator of the numerator of the above to $\displaystyle \sqrt{1 + 2x}$ to become

$\displaystyle =\frac{\frac{(\sqrt{1 + 2x})(-2x)(\sqrt{1 + 2x}) - (1 - x^2)}{\sqrt{1 + 2x}}}{1 + 2x}$

then, simplifying them, I get

$\displaystyle =\frac{(-2x)(1 + 2x) - (1 - x^2)}{\sqrt{1 + 2x}} \times \frac{1}{1 + 2x}$

Is it against the math logic if I equate the denominator of the fractions in the numerator of the expression?

4. Originally Posted by mark1950

from
$\displaystyle =\frac{\sqrt{1+2x}(-2x) - (1-x^2)\frac{1}{\sqrt{1 + 2x}}}{1 + 2x}$

I equated the denominator of the numerator of the above to $\displaystyle \sqrt{1 + 2x}$ to become

$\displaystyle =\frac{\frac{(\sqrt{1 + 2x})(-2x)(\sqrt{1 + 2x}) - (1 - x^2)}{\sqrt{1 + 2x}}}{1 + 2x}$

then, simplifying them, I get

$\displaystyle =\frac{(-2x)(1 + 2x) - (1 - x^2)}{\sqrt{1 + 2x}} \times \frac{1}{1 + 2x}$

Is it against the math logic if I equate the denominator of the fractions in the numerator of the expression?

HI Mark , you are not wrong . Just that you worked it out wrongly .

For the numerator :

$\displaystyle \sqrt{1+2x}(-2x) - (1-x^2)\frac{1}{\sqrt{1 + 2x}}$

$\displaystyle \sqrt{1+2x}(-2x) - \frac{1-x^2}{\sqrt{1 + 2x}}$

$\displaystyle \frac{(1+2x)(-2x)}{\sqrt{1+2x}} - \frac{1-x^2}{\sqrt{1 + 2x}}$

5. Huh? What's the difference? If I simplify your numerator, wouldn't it be the same as my numerator?

6. Hey, wait...I think I know the problem now! Yes, I mistakenly canceled out the $\displaystyle 1 + 2x$ instead of multiplying it with the denominator which is $\displaystyle \sqrt{1 + 2x}$. Oh...I feel kinda silly now. Not enough practice, I must say. Anyway, thanks for your time!