How to differentiate y=√(1-x) / √(1+x) using quotient rule

I know you need to use the quotient rule, have given it a good go but got stuck.
Cheers :)

Okay, so first we must know what the quotient rule is.

The quotient rule is as follows: ((v)(du/dx) - (u)(dv/dx))/(v)²

First, assign values.

u = √(1 - x)
v = √(1 + x)

Now plug in the values:

((√(1 + x))d/dx(√(1 - x)) - (√(1 - x)d/dx(√(1 + x)))/(√(1 + x))²

The next step I would take is to derive both u and v for use in the formula.

d/dx(√(1 - x)) = d/dx(1 - x)^¹/₂

To derive this, use the chain rule.

y = (u)^¹/₂ → ¹/₂(u)^-¹/₂
u = 1 - x → -x

Plug in u and multiply by the derivative of u.

-¹/₂x(1 - x)^-¹/₂

Simplify.

(-¹/₂x)/(√(1 - x))

Now derive v.

y = (u)^¹/₂ → ¹/₂(u)^-¹/₂
u = 1 + x → x

Plug in u and multiply by the derivative of u.

¹/₂x(1 + x)^-¹/₂

Simplify.

(¹/₂x)/(√(1 + x))

Now that we have derived u and v, plug them into the quotient rule.

((√(1 + x))((-¹/₂x)/(√(1 - x))) - (√(1 - x))((¹/₂x)/(√(1 + x)))/(√(1 + x))²

Now do the math. I get:

(-¹/₂x(1 + x) - ¹/₂x(1 - x))/((√(1 - x))(1 + x))

I like this song:

(low, d-high - high, d-low)/ (low, low)

where:

f(x) = high/low

The d's indicate that we take the derivative of that component.


So if f(x) = (2x^3)/(4x^2)

f'(x) = [(4x^2)•(6x^2) - (2x^3)•(8x)] / [(4x^2)^2]


Just do the same for your problem. Take the derivatives of both top and bottom and apply them where the song tells you to.

Make it as kindergaarden as possible: t'is my view.

Quotient rule: d/dx u/v = v du/dx - u dv/dx / v^2
Now, in order to differentiate your u and your v, you're going to need the chain rule.
Think of √(1-x) as (1-x)^1/2, and differentiate accordingly, doing the same for √(1+x).
Then expand or factorise or do whatever you want to get it to look neat.

u = (1-x)^1/2
du/dx = -1/2(1-x)^-1/2
v = (1+x)^1/2
dv/dx = 1/2(1+x)^-1/2

I'm sure you can work from there.

Where did you get stuck?

dy/dx = [ sqrt(1 + x) * d/dx[sqrt(1-x)] - sqrt(1 - x) * d/dx[sqrt(1+x)] ]/ [sqrt(1+x)]^2

Those two derivatives in the numerator are straightforward. So what did you get and what do you mean by "stuck"?

-(1/(Sqrt[1 - x] (1 + x)^(3/2)))

1-x