# Calculus I - Double Derivitive

1. Oct 27, 2009

### RaptorsFan

1. The problem statement, all variables and given/known data

Find dy/dx and d^2y/dx^2

y = x / (x^(2)+1)

2. Relevant equations

d/dx (f/g) = (g d/dx f - f d/dx g) / g^2

3. The attempt at a solution

Finding d/dx:

d/dx y = (x^(2)+1) d/dx (x) - (x) d/dx (x^(2) + 1) / (x^(2)+1)^2

= (x^2 + 1) - (2x^2) / (x^(2)+1)^2

So that's my first derivative answer.. now on to the second.

d/dx(d/dx y) (x^2 + 1)^2 d/dx [(x^(2)+1)-(2x^2)] / (x^(2)+1)^4

((x^(2)+1)^2)(2x-4x)-[4x(x^(2)+1)-(2x^2)/ (x^(2)+1)^4

So, there is bound to be a mistake somewhere.. thank you in advance

2. Oct 27, 2009

### lanedance

i find it quite difficult to read, as you leave out brackets,

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

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

$$= \frac{((x^2 + 1) - 2x^2) }{(x^2 + 1)^2}$$
looks similar to what you had, but with an extra bracket,

though you should also simplify before differentiating again
$$= \frac{ 1 - x^2 }{(x^2 + 1)^2}$$

as mentioned its hard to read without missing +,-,= & brackets

that said I find wreting it as below, then diifferntiating again using the product rule a little easier, though it will lead to identical result as the quotient rule
$$\frac{dy}{dx} = (1 - x^2)(x^2 + 1)^{-2}$$

Last edited: Oct 27, 2009