Find derivative of the following function?

Question

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

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Answer 1 · 2018-03-19T16:19:03

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.

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

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

$= cos^2 theta - sin^2 theta$

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

$= (-2x)/2 = -x$