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#