Question

Find derivative of the following function?

`cos[2cot^-1sqrt((1-x)/(1+x)) ]dy/dx`

Answer 1

The derivative (w.r.t. `x`) of `cos(cot^-1(sqrt(1-x)/sqrt(1+x)))` is `-1`.

The derivative (w.r.t. `x`) of `cos(cot^-1(sqrt(1-x)/sqrt(1+x)))dy/dx` is shown below.

Explanation:

The derivative (w.r.t. `x`) of `cos(cot^-1(sqrt(1-x)/sqrt(1+x)))dy/dx` requires the product rule and is `-dy/dx-x(d^2y)/(dx^2)`

Note that

`cos(cot^-1(sqrt(1-x)/sqrt(1+x))) = -x" "` (for `-1< x<=1`)

Proof

Let `theta = cot^-1(sqrt(1-x)/sqrt(1+x))`

Then `cot theta = sqrt(1-x)/sqrt(1+x)`

Use whatever trigonometric method you prefer to find

`sin theta = sqrt(1+x)/sqrt2` and `cos theta = sqrt(1-x)/sqrt2`.

Now, we see that

`cos(cot^-1(sqrt(1-x)/sqrt(1+x))) = cos(2theta)`

`= cos^2 theta - sin^2 theta`

`= (1-x)/sqrt2 - (1+x)/sqrt2`

`= (-2x)/2 = -x`